begin
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
func InterchangeLine (
M,
l,
k)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
set ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) &
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
theorem
for
m,
n,
l,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
for
i being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= InterchangeLine (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= Line (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b6 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= Line (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b6 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) &
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= Line (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b6 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b6 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) ) ;
theorem
for
m,
n being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
M being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
. j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) : ( ( ) ( )
set )
= a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
* (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) : ( ( ) ( )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) ;
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
func ScalarXLine (
M,
l,
a)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
set ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= a : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) ))
* (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
theorem
for
n,
m,
l,
i being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= ScalarXLine (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
* (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= Line (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b7 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b7 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) ) ;
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
assume
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
;
func RlineXScalar (
M,
l,
k,
a)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= (a : ( ( ) ( ) set ) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
+ (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,j : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
theorem
for
n,
m,
l,
k,
i being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= RlineXScalar (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) holds
( (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b9 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= (a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) * (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) Function-like V35( width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like ) Element of (width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( V21() V22() V23() V27() ext-real non negative ) M2( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) -tuples_on the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) : ( ( ) ( functional V11() FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
+ (Line (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) & (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) implies
Line (
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b9 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b9 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= Line (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
Function-like V35(
width b8 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) )
FinSequence-like )
Element of
(width b8 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b6 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
-tuples_on the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) : ( ( ) (
functional V11()
FinSequence-membered )
FinSequenceSet of the
carrier of
b6 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) ) ) ;
notation
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
synonym RLineXS (
M,
l,
k,
a)
for RlineXScalar (
M,
l,
k,
a);
end;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
(ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= ILine (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
n being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
l being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
(SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b2 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= SXLine (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b2 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
(RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= RLineXS (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
n,
m,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
M being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) holds
ILine (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
n,
m,
l,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
M being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) holds
ILine (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= ILine (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
n,
m,
l,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
M being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
ILine (
(ILine (M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b1 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b1 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
(
ILine (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) is
invertible &
(ILine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= ILine (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) holds
(
RLineXS (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) is
invertible &
(RLineXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= RLineXS (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
(- a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( )
M2( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ) ;
theorem
for
l,
n being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
<> 0. K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V47(
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ) )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) holds
(
SXLine (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) is
invertible &
(SXLine ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= SXLine (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ") : ( ( ) ( )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ) ;
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
assume that
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
and
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
and
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
and
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
;
func InterchangeCol (
M,
l,
k)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
set ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> k : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
assume that
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
and
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
and
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
;
func ScalarXCol (
M,
l,
a)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
set ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= a : ( (
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) ) (
V1()
V4(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
V5(
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
Function-like V18(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ,
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ) )
M2(
bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) ))
* (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
set )
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
definition
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
assume that
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
and
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,m : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
and
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
and
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
;
func RcolXScalar (
M,
l,
k,
a)
-> ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( ( ) ( )
1-sorted ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) , ( ( ) ( )
set ) )
means
(
len it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) ))
= len M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) & ( for
i,
j being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) st
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) holds
( (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
= l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= (a : ( ( ) ( ) set ) * (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) ( V1() V4(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) V5(K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) Function-like V18(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) ) M2( bool [:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( ) M2( the carrier of K : ( ( Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
+ (M : ( ( ) ( ) M2(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) )) * (i : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,l : ( ( Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) ( V1() V4([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ) V5(m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) Function-like V18([: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ) ) M2( bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) & (
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[: the carrier of n : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) implies
it : ( ( ) ( )
M2(
bool M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )) : ( ( ) ( )
set ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) ))
= M : ( ( ) ( )
M2(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ))
* (
i : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
j : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) ( )
M2( the
carrier of
K : ( (
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) ) (
V1()
V4(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) )
V5(
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) )
Function-like V18(
[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) ,
m : ( ( ) ( )
VectSpStr over
n : ( ( ) ( )
1-sorted ) ) ) )
M2(
bool [:[:m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( ) set ) ,m : ( ( ) ( ) VectSpStr over n : ( ( ) ( ) 1-sorted ) ) :] : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) ) ) ) );
end;
notation
let n,
m be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let K be ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ;
let M be ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
let l,
k be ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ;
let a be ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ;
synonym RColXS (
M,
l,
k,
a)
for RcolXScalar (
M,
l,
k,
a);
end;
theorem
for
l,
k,
n,
m being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) holds
(ILine (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= ICol (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
m being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) holds
(SXLine (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= SXCol (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
k,
n,
m being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
M,
M1 being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in Seg (width M : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
M2(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
m : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) &
M1 : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
= M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ) holds
(RLineXS (M1 : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b5 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b3 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b4 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
@ : ( (
tabular ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
FinSequence of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
= RColXS (
M : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b5 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b3 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b4 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* (ICol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= ICol (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* (SXCol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= SXCol (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
for
A being ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
* (RColXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= RColXS (
A : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
(ICol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= ICol (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n,
k being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
<> l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
(RColXS ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b4 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,k : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= RColXS (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
k : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
(- a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ) : ( ( ) ( )
M2( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b4 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;
theorem
for
l,
n being ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
for
K being ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field)
for
a being ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) ) st
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
in dom (1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) : ( ( ) ( )
M2(
bool NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) : ( ( ) ( )
set ) )) &
a : ( ( ) ( )
Element of ( ( ) (
V11() non
trivial )
set ) )
<> 0. K : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V47(
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) ) )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) &
n : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat)
> 0 : ( ( ) (
V11()
V21()
V22()
V23()
V25()
V26()
V27()
ext-real non
positive non
negative )
M3(
REAL : ( ( ) ( )
set ) ,
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )) holds
(SXCol ((1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) ( V1() V4( NAT : ( ( ) ( V11() V21() V22() V23() ) M2( bool REAL : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ) V5( the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) * : ( ( ) ( functional FinSequence-membered ) FinSequenceSet of the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ) Function-like FinSequence-like tabular ) Matrix of b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,b2 : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) , the carrier of b3 : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) : ( ( ) ( V11() non trivial ) set ) ) ,l : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) ,a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) )
~ : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )
= SXCol (
(1. (K : ( ( V40() non degenerated right_complementable almost_left_invertible V113() V115() V118() V119() V120() well-unital V132() ) ( V40() non degenerated non trivial right_complementable almost_left_invertible unital V113() V115() V118() V119() V120() right-distributive left-distributive right_unital well-unital V132() left_unital V182() ) Field) ,n : ( ( V27() ) ( V21() V22() V23() V27() ext-real non negative ) Nat) )) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) ,
l : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
(a : ( ( ) ( ) Element of ( ( ) ( V11() non trivial ) set ) ) ") : ( ( ) ( )
M2( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) )) ) : ( ( ) (
V1()
V4(
NAT : ( ( ) (
V11()
V21()
V22()
V23() )
M2(
bool REAL : ( ( ) ( )
set ) : ( ( ) ( )
set ) )) )
V5( the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set )
* : ( ( ) (
functional FinSequence-membered )
FinSequenceSet of the
carrier of
b3 : ( (
V40() non
degenerated right_complementable almost_left_invertible V113()
V115()
V118()
V119()
V120()
well-unital V132() ) (
V40() non
degenerated non
trivial right_complementable almost_left_invertible unital V113()
V115()
V118()
V119()
V120()
right-distributive left-distributive right_unital well-unital V132()
left_unital V182() )
Field) : ( ( ) (
V11() non
trivial )
set ) ) )
Function-like FinSequence-like tabular )
Matrix of
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) ,
b2 : ( (
V27() ) (
V21()
V22()
V23()
V27()
ext-real non
negative )
Nat) , ( ( ) (
V11() non
trivial )
set ) ) ;