begin
theorem
for
x being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x : ( ( ) ( )
Variable)
in variables_in H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) iff
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
x : ( ( ) ( )
Variable) ,
H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) ;
theorem
for
x being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
m being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x : ( ( ) ( )
Variable)
in variables_in H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) iff
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
m : ( ( ) ( )
Element of
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ) ;
theorem
for
x,
y,
z being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
m1,
m2,
m3 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) )
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
x : ( ( ) ( )
Variable)
<> y : ( ( ) ( )
Variable) &
y : ( ( ) ( )
Variable)
<> z : ( ( ) ( )
Variable) &
z : ( ( ) ( )
Variable)
<> x : ( ( ) ( )
Variable) holds
(
((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) / (x : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (y : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
z : ( ( ) ( )
Variable) ,
m3 : ( ( ) ( )
Element of
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= ((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) / (z : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (y : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
m1 : ( ( ) ( )
Element of
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) / (x : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (y : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
z : ( ( ) ( )
Variable) ,
m3 : ( ( ) ( )
Element of
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= ((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) / (y : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b4 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (z : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b4 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
m1 : ( ( ) ( )
Element of
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b4 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b4 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
x1,
x2,
x3,
x4 being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
m1,
m2,
m3,
m4 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) )
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
x1 : ( ( ) ( )
Variable)
<> x2 : ( ( ) ( )
Variable) &
x1 : ( ( ) ( )
Variable)
<> x3 : ( ( ) ( )
Variable) &
x1 : ( ( ) ( )
Variable)
<> x4 : ( ( ) ( )
Variable) &
x2 : ( ( ) ( )
Variable)
<> x3 : ( ( ) ( )
Variable) &
x2 : ( ( ) ( )
Variable)
<> x4 : ( ( ) ( )
Variable) &
x3 : ( ( ) ( )
Variable)
<> x4 : ( ( ) ( )
Variable) holds
(
(((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x4 : ( ( ) ( )
Variable) ,
m4 : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x4 : ( ( ) ( ) Variable) ,m4 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x1 : ( ( ) ( )
Variable) ,
m1 : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
(((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
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Element of
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b5 : ( ( non
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Element of
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b5 : ( ( non
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b5 : ( ( non
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Element of
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b5 : ( ( non
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Element of
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b5 : ( ( non
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= (((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x4 : ( ( ) ( ) Variable) ,m4 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
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Element of
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b5 : ( ( non
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Relation-like VAR : ( ( ) ( non
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Element of
K27(
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Element of
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Element of
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V30(
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Element of
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b5 : ( ( non
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b5 : ( ( non
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x2 : ( ( ) ( )
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b5 : ( ( non
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Element of
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b5 : ( ( non
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Element of
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Element of
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V30(
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Element of
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b5 : ( ( non
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Element of
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Element of
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Element of
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b5 : ( ( non
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(((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
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Element of
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b5 : ( ( non
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Element of
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Element of
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V30(
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Element of
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b5 : ( ( non
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Element of
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Element of
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b5 : ( ( non
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x4 : ( ( ) ( )
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m4 : ( ( ) ( )
Element of
b5 : ( ( non
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Function-like V30(
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Element of
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Element of
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b5 : ( ( non
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Relation-like VAR : ( ( ) ( non
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Element of
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Element of
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REAL : ( ( ) ( )
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-defined b5 : ( ( non
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VAR : ( ( ) ( non
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Element of
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Element of
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V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x4 : ( ( ) ( ) Variable) ,m4 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x1 : ( ( ) ( )
Variable) ,
m1 : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
x1,
x2,
x3,
x4 being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
m1,
m2,
m,
m3,
m4 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) )
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) holds
(
((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x1 : ( ( ) ( )
Variable) ,
m : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x1 : ( ( ) ( )
Variable) ,
m : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) &
(((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
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= ((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
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b5 : ( ( non
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b5 : ( ( non
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((((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x1 : ( ( ) ( ) Variable) ,m1 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x4 : ( ( ) ( ) Variable) ,m4 : ( ( ) ( ) Element of b5 : ( ( non 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Element of
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Element of
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Element of
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Element of
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Element of
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Element of
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Element of
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V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= (((v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) / (x2 : ( ( ) ( ) Variable) ,m2 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x3 : ( ( ) ( ) Variable) ,m3 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) ) Element of K27(K28(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b5 : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) / (x4 : ( ( ) ( ) Variable) ,m4 : ( ( ) ( ) Element of b5 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
x1 : ( ( ) ( )
Variable) ,
m : ( ( ) ( )
Element of
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b5 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b5 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
x being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
m being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x : ( ( ) ( )
Variable)
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) iff
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
m : ( ( ) ( )
Element of
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b2 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b2 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) holds
for
m1,
m2 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) ) holds
(
(def_func' (H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) ,v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. m1 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= m2 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) iff
M : ( ( non
empty ) ( non
empty )
set ) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) / ((x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,m1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
(x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
m2 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
c= {(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set ) &
M : ( ( non
empty ) ( non
empty )
set )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) holds
def_func' (
H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= def_func (
H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
M : ( ( non
empty ) ( non
empty )
set ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ;
theorem
for
x,
y being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x : ( ( ) ( )
Variable)
in variables_in H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
/ (
y : ( ( ) ( )
Variable) ,
x : ( ( ) ( )
Variable) ) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) iff
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
/ (
y : ( ( ) ( )
Variable) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . x : ( ( ) ( ) Variable) ) : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ) ;
theorem
for
x,
y being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x : ( ( ) ( )
Variable)
in variables_in H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) holds
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) Variable) ) : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
/ (
y : ( ( ) ( )
Variable) ,
x : ( ( ) ( )
Variable) ) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ;
theorem
for
x,
y being ( ( ) ( )
Variable)
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) & not
x : ( ( ) ( )
Variable)
in variables_in H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) & not
y : ( ( ) ( )
Variable)
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
x : ( ( ) ( )
Variable)
<> x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) &
x : ( ( ) ( )
Variable)
<> x. 3 : ( ( ) ( non
empty ext-real positive non
negative V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) &
x : ( ( ) ( )
Variable)
<> x. 4 : ( ( ) ( non
empty ext-real positive non
negative V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) holds
( not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free (H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) / (y : ( ( ) ( ) Variable) ,x : ( ( ) ( ) Variable) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
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set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) )
/ (
x : ( ( ) ( )
Variable) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) Variable) ) : ( ( ) ( )
Element of
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,((H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) / (y : ( ( ) ( ) Variable) ,x : ( ( ) ( ) Variable) )) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) &
def_func' (
H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b3 : ( ( non
empty ) ( non
empty )
set )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= def_func' (
(H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) / (y : ( ( ) ( ) Variable) ,x : ( ( ) ( ) Variable) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) / (x : ( ( ) ( ) Variable) ,(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b3 : ( ( non empty ) ( non empty ) set ) ) . y : ( ( ) ( ) Variable) ) : ( ( ) ( ) Element of b3 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( (
Function-like V30(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b3 : ( ( non
empty ) ( non
empty )
set )
-defined b3 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b3 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b3 : ( ( non
empty ) ( non
empty )
set ) ,
b3 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
i being ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
for
H1 being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v1 being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H1 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v1 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H1 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) holds
ex
H2 being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ex
v2 being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
( ( for
j being ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) st
j : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
< i : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) &
x. j : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in variables_in H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( non
empty ) ( non
empty finite )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) & not
j : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
= 3 : ( ( ) ( non
empty ext-real positive non
negative V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) holds
j : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )
= 4 : ( ( ) ( non
empty ext-real positive non
negative V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ) & not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v2 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H2 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) &
def_func' (
H1 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v1 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= def_func' (
H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v2 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H1 being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v1 being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H1 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v1 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H1 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) holds
ex
H2 being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ex
v2 being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
(
(Free H1 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) ) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
/\ (Free H2 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) ) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
c= {(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set ) & not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v2 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H2 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) &
def_func' (
H1 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v1 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
= def_func' (
H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) ,
v2 : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st not
x. 0 : ( ( ) (
ext-real V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) iff for
m1 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) ) ex
m2 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) ) st
for
m3 being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) ) holds
(
M : ( ( non
empty ) ( non
empty )
set ) ,
(v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) / ((x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,m1 : ( ( ) ( ) Element of b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
/ (
(x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
m3 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) iff
m3 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) )
= m2 : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
F,
G being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
F : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
is_definable_in M : ( ( non
empty ) ( non
empty )
set ) &
G : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
is_definable_in M : ( ( non
empty ) ( non
empty )
set ) &
Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
c= {(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set ) holds
for
FG being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
dom FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= M : ( ( non
empty ) ( non
empty )
set ) & ( for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) holds
( (
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) implies
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. (v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set )
= F : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. (v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) & (
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= 'not' H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) implies
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. (v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set )
= G : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. (v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) . (x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) ) ) holds
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
is_definable_in M : ( ( non
empty ) ( non
empty )
set ) ;
theorem
for
M being ( ( non
empty ) ( non
empty )
set )
for
H1,
H2,
H being ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula)
for
v being ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
M : ( ( non
empty ) ( non
empty )
set ) ) st
{(x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 1 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 2 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set )
misses Free H1 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H1 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) &
{(x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 1 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 2 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set )
misses Free H2 : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) &
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
|= All (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
(Ex ((x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(All ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(H2 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) <=> ((x. 4 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) '=' (x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) ) : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) M7( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) )) : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) ) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) &
{(x. 0 : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 1 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) ,(x. 2 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( ) Element of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) } : ( ( ) (
finite )
set )
misses Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) & not
x. 4 : ( ( ) ( non
empty ext-real positive non
negative V18()
V19()
V20()
V21()
V25()
V26() )
Element of
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
in Free H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) ( )
Element of
K27(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) holds
for
FG being ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) st
dom FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function) : ( ( ) ( )
set )
= M : ( ( non
empty ) ( non
empty )
set ) & ( for
m being ( ( ) ( )
Element of
M : ( ( non
empty ) ( non
empty )
set ) ) holds
( (
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
/ (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) implies
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set )
= (def_func' (H1 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) ,v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) & (
M : ( ( non
empty ) ( non
empty )
set ) ,
v : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Function of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) )
/ (
(x. 3 : ( ( ) ( non empty ext-real positive non negative V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( )
Element of
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ) ,
m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) : ( (
Function-like V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) )
V30(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
VAR : ( ( ) ( non
empty )
Element of
K27(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) ) : ( ( ) ( )
set ) ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
|= 'not' H : ( (
ZF-formula-like ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
ZF-formula) : ( ( ) (
Relation-like NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-defined NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
-valued Function-like finite V41()
V42()
ZF-formula-like )
M7(
NAT : ( ( ) ( non
empty V19()
V20()
V21() )
Element of
K27(
REAL : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) ) )) implies
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
. m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set )
= (def_func' (H2 : ( ( ZF-formula-like ) ( Relation-like NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like finite V41() V42() ZF-formula-like ) ZF-formula) ,v : ( ( Function-like V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like non empty V14( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) V30( VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) ) Function of VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ,b1 : ( ( non empty ) ( non empty ) set ) ) )) : ( (
Function-like V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) (
Relation-like b1 : ( ( non
empty ) ( non
empty )
set )
-defined b1 : ( ( non
empty ) ( non
empty )
set )
-valued Function-like non
empty V14(
b1 : ( ( non
empty ) ( non
empty )
set ) )
V30(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) )
Element of
K27(
K28(
b1 : ( ( non
empty ) ( non
empty )
set ) ,
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
set ) ) : ( ( ) ( )
set ) )
. m : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) : ( ( ) ( )
Element of
b1 : ( ( non
empty ) ( non
empty )
set ) ) ) ) ) holds
FG : ( (
Relation-like Function-like ) (
Relation-like Function-like )
Function)
is_parametrically_definable_in M : ( ( non
empty ) ( non
empty )
set ) ;