begin
registration
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let L be ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let p,
q be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero non
trivial )
set ) ) ;
cluster p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
*' q : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-> Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero ;
end;
begin
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
- (p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) + q : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= (- p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
+ (- q : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable Abelian add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable Abelian add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(0_ (X : ( ( ) ( ) set ) ,L : ( ( non empty left_zeroed ) ( non empty left_zeroed ) addLoopStr ) )) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_zeroed ) ( non empty left_zeroed ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
+ p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_zeroed ) ( non empty left_zeroed ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_zeroed ) ( non
empty left_zeroed )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(
(- p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
+ p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= 0_ (
X : ( ( ) ( )
set ) ,
L : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
+ (- p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= 0_ (
X : ( ( ) ( )
set ) ,
L : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( ( ) ( )
set )
for
L being ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
- (0_ (n : ( ( ) ( ) set ) ,L : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) )) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(0_ (n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,L : ( ( non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
*' p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable left-distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
(
- (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' q : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= (- p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
*' q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
- (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' q : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' (- q : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
(m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
. ((term m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) set ) + b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) ) : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
set ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
= (m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) . (term m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) set ) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
* (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) . b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(0. L : ( ( non empty left_add-cancelable left-distributive right_zeroed ) ( non empty left_add-cancelable left-distributive right_zeroed ) doubleLoopStr ) ) : ( ( ) (
zero left_add-cancelable )
Element of the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
* p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_add-cancelable left-distributive right_zeroed ) ( non empty left_add-cancelable left-distributive right_zeroed ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= 0_ (
X : ( ( ) ( )
set ) ,
L : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left_add-cancelable left-distributive right_zeroed ) ( non
empty left_add-cancelable left-distributive right_zeroed )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left_add-cancelable left-distributive right_zeroed ) ( non empty left_add-cancelable left-distributive right_zeroed ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
for
a being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) holds
(
- (a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= (- a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero ) set ) ) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
* p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
- (a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= a : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
* (- p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
for
a,
a9 being ( ( ) ( )
Element of ( ( ) ( non
zero )
set ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
+ (a9 : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= (a : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) + a9 : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) )
* p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty left-distributive ) ( non
empty left-distributive )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty left-distributive ) ( non empty left-distributive ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
X being ( ( ) ( )
set )
for
L being ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 )
for
p being ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags X : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
for
a,
a9 being ( ( ) ( )
Element of ( ( ) ( non
zero )
set ) ) holds
(a : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) * a9 : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) ) : ( ( ) ( )
Element of the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) )
* p : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= a : ( ( ) ( )
Element of ( ( ) ( non
zero )
set ) )
* (a9 : ( ( ) ( ) Element of ( ( ) ( non zero ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ) ( ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( ( ) ( )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty associative ) ( non
empty associative )
multLoopStr_0 ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ) ( ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty associative ) ( non empty associative ) multLoopStr_0 ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p,
p9 being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
for
a being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) holds
a : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
* (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) *' p9 : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
*' (a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero ) set ) ) * p9 : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed ) ( non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
begin
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let b be ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ;
let L be ( ( non
empty ) ( non
empty )
ZeroStr ) ;
let p be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) ;
func b *' p -> ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( )
set ) )
means
for
b9 being ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ) st
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
divides b9 : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
(
it : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
. b9 : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) : ( ( ) ( )
Element of the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) )
= p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
. (b9 : ( ( Relation-like n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) -' b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total )
set ) : ( ( ) ( )
Element of the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) ) & ( for
b9 being ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ) st not
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
divides b9 : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
it : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
. b9 : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) : ( ( ) ( )
Element of the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) )
= 0. L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
Element of the
carrier of
L : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
set ) ) ) );
end;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
b,
b9 being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b4 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b4 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
. (b9 : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) + b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) ) : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
set ) : ( ( ) ( )
Element of the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
. b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) : ( ( ) ( )
Element of the
carrier of
b4 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) )
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
Support (b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non empty ) ( non empty ) ZeroStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
c= { (b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) + b9 : ( ( ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) Element of Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) ) : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) set ) where b9 is ( ( ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) Element of Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) : b9 : ( ( ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) Element of Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) in Support p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non empty ) ( non empty ) ZeroStr ) ) : ( ( ) ( functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) } ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) )
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
HT (
(b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero non
trivial )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
= b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
+ (HT (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ,T : ( ( total reflexive antisymmetric connected transitive admissible ) ( Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -valued total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) reflexive antisymmetric connected transitive admissible ) TermOrder of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) )) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
set ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) )
for
b,
b9 being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support (b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non empty ) ( non empty ) ZeroStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
<= b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
+ (HT (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non empty ) ( non empty ) ZeroStr ) ) ,T : ( ( total reflexive antisymmetric connected transitive admissible ) ( Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -valued total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) reflexive antisymmetric connected transitive admissible ) TermOrder of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) )) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
set ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) holds
(EmptyBag n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
*' p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
for
b1,
b2 being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) holds
(b1 : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) + b2 : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) ) : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
set )
*' p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) )
= b1 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
*' (b2 : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
a being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero non
trivial )
set ) ) holds
Support (a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero non trivial ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
c= Support p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) )
for
a being ( ( non
zero ) ( non
zero )
Element of ( ( ) ( non
zero non
trivial )
set ) ) holds
Support p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
c= Support (a : ( ( non zero ) ( non zero ) Element of ( ( ) ( non zero non trivial ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial domRing-like ) ( non empty non trivial domRing-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial domRing-like ) ( non empty non trivial domRing-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial domRing-like ) ( non empty non trivial domRing-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b2 : ( ( non trivial domRing-like ) ( non empty non trivial domRing-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial domRing-like ) ( non
empty non
trivial domRing-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial domRing-like ) ( non empty non trivial domRing-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
a being ( ( non
zero ) ( non
zero left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero non
trivial )
set ) ) holds
HT (
(a : ( ( non zero ) ( non zero left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero non trivial ) set ) ) * p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
= HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable distributive add-associative right_zeroed domRing-like ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
L being ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero non
trivial )
set ) )
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
for
a being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero non
trivial )
set ) ) holds
a : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero non
trivial )
set ) )
* (b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero non trivial ) set ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero non
trivial )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= (Monom (a : ( ( ) ( left_add-cancelable right_add-cancelable add-cancelable right_complementable ) Element of ( ( ) ( non zero non trivial ) set ) ) ,b : ( ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) )) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
*' p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Series of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , ( ( ) ( non
zero non
trivial )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b2 : ( ( non
trivial right_complementable distributive add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b2 : ( ( non trivial right_complementable distributive add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
in Support q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
HT (
(m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
in Support (m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' q : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102() right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
begin
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
empty ) ( non
empty )
ZeroStr ) ;
let p,
q be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ;
pred p <= q,
T means
[(Support p : ( ( Function-like V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Element of bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,(Support q : ( ( Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) ( Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) -defined T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) -valued Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) Element of bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ] : ( ( ) (
V21() )
set )
in FinOrd RelStr(#
(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) #) : ( (
strict ) (
strict )
RelStr ) : ( (
total reflexive antisymmetric transitive ) (
Relation-like Fin the
carrier of
RelStr(#
(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) #) : ( (
strict ) (
strict )
RelStr ) : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
zero cup-closed diff-closed preBoolean )
set )
-defined Fin the
carrier of
RelStr(#
(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) #) : ( (
strict ) (
strict )
RelStr ) : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
zero cup-closed diff-closed preBoolean )
set )
-valued total V46(
Fin the
carrier of
RelStr(#
(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) #) : ( (
strict ) (
strict )
RelStr ) : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
zero cup-closed diff-closed preBoolean )
set ) ,
Fin the
carrier of
RelStr(#
(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) #) : ( (
strict ) (
strict )
RelStr ) : ( ( ) ( )
set ) : ( (
preBoolean ) ( non
zero cup-closed diff-closed preBoolean )
set ) )
reflexive antisymmetric transitive )
Element of
bool [:(Fin the carrier of RelStr(# (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) #) : ( ( strict ) ( strict ) RelStr ) : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non zero cup-closed diff-closed preBoolean ) set ) ,(Fin the carrier of RelStr(# (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) #) : ( ( strict ) ( strict ) RelStr ) : ( ( ) ( ) set ) ) : ( ( preBoolean ) ( non zero cup-closed diff-closed preBoolean ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
empty ) ( non
empty )
ZeroStr ) ;
let p,
q be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ;
pred p < q,
T means
(
p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<= q : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) &
Support p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<> Support q : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) );
end;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) holds
PosetMax (Support (p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ,T : ( ( total reflexive antisymmetric connected transitive ) ( Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -valued total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) reflexive antisymmetric connected transitive ) TermOrder of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) )) : ( ( ) (
finite )
Element of
Fin the
carrier of
RelStr(#
(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) #) : ( (
strict ) ( non
empty strict total reflexive transitive antisymmetric connected )
RelStr ) : ( ( ) ( non
zero )
set ) : ( (
preBoolean ) ( non
zero cup-closed diff-closed preBoolean )
set ) ) : ( ( ) ( )
Element of the
carrier of
RelStr(#
(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b2 : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) #) : ( (
strict ) ( non
empty strict total reflexive transitive antisymmetric connected )
RelStr ) : ( ( ) ( non
zero )
set ) )
= HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
addLoopStr ) ) holds
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
addLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
addLoopStr ) ) holds
( (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) &
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) iff
Support p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= Support q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) : ( ( ) (
functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
addLoopStr )
for
p,
q,
r being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
addLoopStr ) ) st
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) &
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= r : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= r : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
addLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
addLoopStr ) ) holds
(
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) or
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
addLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
addLoopStr ) ) holds
(
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
<= q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) iff not
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) )
< p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty ) ( non
empty )
ZeroStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) holds
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( non
empty ) ( non
empty )
Subset of ) ex
p being ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
(
p : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( ( non
empty ) ( non
empty )
Subset of ) & ( for
q being ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
q : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( ( non
empty ) ( non
empty )
Subset of ) holds
p : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
<= q : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable V102()
right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p,
q being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) holds
(
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) )
< q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) iff ( (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) )
= 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) )
<> 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) or
HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
< HT (
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) or (
HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
= HT (
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) &
Red (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
< Red (
q : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) holds
Red (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
< HM (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) holds
HM (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<= p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) holds
Red (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable add-associative right_zeroed ) ( non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like ) addLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
< p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable add-associative right_zeroed ) ( non
empty non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
addLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
begin
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
p,
g be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
let b be ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ;
pred f reduces_to g,
p,
b,
T means
(
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<> 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
p : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
<> 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
b : ( (
Function-like V46(
[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
in Support f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) & ex
s being ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ) st
(
s : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
+ (HT (p : ( ( Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) ( Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) -defined T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) -valued Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) Element of bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) )) : ( ( ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total )
set )
= b : ( (
Function-like V46(
[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
g : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
- (((f : ( ( Function-like V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) ( non zero Relation-like Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) finite-Support ) Element of bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) . b : ( ( Function-like V46([:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) ) ) ( Relation-like [:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) -defined L : ( ( non empty ) ( non empty ) ZeroStr ) -valued Function-like V46([:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) ) ) Element of bool [:[:L : ( ( non empty ) ( non empty ) ZeroStr ) , the carrier of T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) / (HC (p : ( ( Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) ( Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) -defined T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) -valued Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) Element of bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) )) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) : ( ( ) ( ) Element of the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) * (s : ( ( Relation-like n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined Function-like total V245() finite-support ) ( Relation-like n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) -defined RAT : ( ( ) ( ) set ) -valued Function-like total V242() V243() V244() V245() finite-support ) bag of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) *' p : ( ( Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) ( Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) -defined T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) -valued Function-like V46([: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) ) ) Element of bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ) : ( ( Function-like V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) ( non zero Relation-like Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) -valued Function-like total V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) ) ) Series of ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , ( ( ) ( non zero ) set ) ) ) : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) ) )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) );
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
p,
g be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
pred f reduces_to g,
p,
T means
ex
b being ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) ) st
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to g : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
p : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b : ( (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
g be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
let P be ( ( ) ( )
Subset of ) ;
pred f reduces_to g,
P,
T means
ex
p being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) st
(
p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to g : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) );
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
p be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
pred f is_reducible_wrt p,
T means
ex
g being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) st
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to g : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
let P be ( ( ) ( )
Subset of ) ;
pred f is_reducible_wrt P,
T means
ex
g being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) st
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to g : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
P : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
p,
g be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
pred f top_reduces_to g,
p,
T means
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to g : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined L : ( ( non
empty ) ( non
empty )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
p : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
HT (
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) : ( ( ) (
Relation-like n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set )
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f,
p be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
pred f is_top_reducible_wrt p,
T means
ex
g being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) st
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
top_reduces_to g : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ;
end;
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let f be ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ;
let P be ( ( ) ( )
Subset of ) ;
pred f is_top_reducible_wrt P,
T means
ex
p being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) st
(
p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
f : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
is_top_reducible_wrt p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) );
end;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
(
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
is_reducible_wrt p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) iff ex
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
(
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) &
HT (
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
divides b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
is_irreducible_wrt p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- (m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
HT (
(m : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) non-zero monomial-like finite-Support ) Monomial of b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) *' p : ( ( Function-like V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
in Support f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
not
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
b,
b9 being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
< b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) &
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
(
b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) iff
b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
b,
b9 being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
< b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) &
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
. b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
= g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
. b9 : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
for
b being ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ) st
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
in Support g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( ( ) (
functional finite )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
b : ( (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined Function-like total V245()
finite-support ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
bag of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) )
<= HT (
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
p,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
p : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
< f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
begin
definition
let n be ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ;
let T be ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
let L be ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ;
let P be ( ( ) ( )
Subset of ) ;
func PolyRedRel (
P,
T)
-> ( ( ) (
Relation-like NonZero (Polynom-Ring (n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) )) : ( ( non
empty strict ) ( non
empty strict )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) )) : ( ( non
empty strict ) ( non
empty strict )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ,L : ( ( non empty ) ( non empty ) ZeroStr ) )) : ( ( non
empty strict ) ( non
empty strict )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
means
for
p,
q being ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty ) ( non
empty )
ZeroStr ) ) holds
(
[p : ( ( Function-like V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ,q : ( ( Function-like V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) ( non zero Relation-like Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) -defined the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) -valued Function-like total V46( Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) ) finite-Support ) Polynomial of ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) ,L : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) ) ] : ( ( ) (
V21() )
set )
in it : ( (
Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) ) (
Relation-like [: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set )
-defined T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr )
-valued Function-like V46(
[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) ) )
Element of
bool [:[: the carrier of n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) : ( ( ) ( ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) ( Relation-like ) set ) ,T : ( ( non trivial ) ( non empty non trivial ) ZeroStr ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) iff
p : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to q : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
P : ( (
Function-like V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support ) ( non
zero Relation-like Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set )
-valued Function-like total V46(
Bags n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
set ) : ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
L : ( ( non
empty ) ( non
empty )
ZeroStr ) : ( ( ) ( non
zero )
set ) )
finite-Support )
Element of
bool [:(Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero functional ) Element of bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) set ) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of L : ( ( non empty ) ( non empty ) ZeroStr ) : ( ( ) ( non zero ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
T : ( ( non
trivial ) ( non
empty non
trivial )
ZeroStr ) );
end;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
P being ( ( ) ( )
Subset of ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
(
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
<= f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) & (
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
= 0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) or
HT (
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
<= HT (
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
-defined RAT : ( ( ) ( )
set )
-valued Function-like total V242()
V243()
V244()
V245()
finite-support )
Element of
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) ) ;
theorem
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
h being ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( ( ) ( )
Subset of ) holds
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued co-well_founded weakly-normalizing strongly-normalizing irreflexive )
Relation of ,)
reduces h : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' f : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
0_ (
n : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
reduces_to g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) holds
m : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Monomial of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
reduces_to m : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
non-zero monomial-like finite-Support )
Monomial of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces m : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Monomial of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
m : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Monomial of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
for
m being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Monomial of
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces m : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like finite-Support )
Monomial of
b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
*' f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g,
h,
h1 being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= h : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) &
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces h : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
h1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
ex
f1,
g1 being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
(
f1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- g1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
= h1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) &
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
f1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) &
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g1 : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
are_convergent_wrt PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
are_convertible_wrt PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) st
f : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) ,
g : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
are_convertible_wrt PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,) holds
f : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) ,
g : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
are_congruent_mod P : ( ( ) ( )
Subset of )
-Ideal : ( ( non
zero add-closed left-ideal right-ideal ) ( non
zero add-closed left-ideal right-ideal )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat)
for
T being ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
empty non
degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( non
empty ) ( non
empty )
Subset of )
for
f,
g being ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) st
f : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) ,
g : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
are_congruent_mod P : ( ( non
empty ) ( non
empty )
Subset of )
-Ideal : ( ( non
zero add-closed left-ideal right-ideal ) ( non
zero add-closed left-ideal right-ideal )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
f : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) ) ,
g : ( ( ) (
left_add-cancelable right_add-cancelable add-cancelable right_complementable )
Element of ( ( ) ( non
zero )
set ) )
are_convertible_wrt PolyRedRel (
P : ( ( non
empty ) ( non
empty )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive admissible ) (
Relation-like Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
natural ) (
epsilon-transitive epsilon-connected ordinal natural finite cardinal V49()
ext-real non
negative complex )
Nat) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive well_founded admissible )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural finite cardinal V49() ext-real non negative complex ) Nat) ,b3 : ( ( non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued co-well_founded weakly-normalizing strongly-normalizing irreflexive )
Relation of ,) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f,
g being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) holds
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
- g : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
in P : ( ( ) ( )
Subset of )
-Ideal : ( ( non
zero add-closed left-ideal right-ideal ) ( non
zero add-closed left-ideal right-ideal )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;
theorem
for
n being ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal)
for
T being ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
for
L being ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr )
for
P being ( ( ) ( )
Subset of )
for
f being ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags n : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) st
PolyRedRel (
P : ( ( ) ( )
Subset of ) ,
T : ( (
total reflexive antisymmetric connected transitive ) (
Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-valued total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) )
reflexive antisymmetric connected transitive )
TermOrder of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ) ) : ( ( ) (
Relation-like NonZero (Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set )
-valued )
Relation of ,)
reduces f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) ,
0_ (
n : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) ,
L : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) ) : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) ) ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
monomial-like Constant finite-Support )
Element of
bool [:(Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero functional ) Element of bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non zero ) set ) : ( ( ) ( cup-closed diff-closed preBoolean ) set ) ) , the carrier of b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) : ( ( ) ( non zero non trivial ) set ) :] : ( ( ) (
Relation-like )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) holds
f : ( (
Function-like V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support ) ( non
zero Relation-like Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) )
-defined the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set )
-valued Function-like total V46(
Bags b1 : ( (
ordinal ) (
epsilon-transitive epsilon-connected ordinal )
Ordinal) : ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) , the
carrier of
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero non
trivial )
set ) )
finite-Support )
Polynomial of ( ( ) ( non
zero functional )
Element of
bool (Bags b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ,
b3 : ( ( non
trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non
empty non
degenerated non
trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) )
in P : ( ( ) ( )
Subset of )
-Ideal : ( ( non
zero add-closed left-ideal right-ideal ) ( non
zero add-closed left-ideal right-ideal )
Element of
bool the
carrier of
(Polynom-Ring (b1 : ( ( ordinal ) ( epsilon-transitive epsilon-connected ordinal ) Ordinal) ,b3 : ( ( non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed ) ( non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like ) doubleLoopStr ) )) : ( ( non
empty strict ) ( non
empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict V102()
associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like )
doubleLoopStr ) : ( ( ) ( non
zero )
set ) : ( ( ) (
cup-closed diff-closed preBoolean )
set ) ) ;