:: ZFMODEL2 semantic presentation

begin

theorem :: ZFMODEL2:1
for x, y being ( ( ) ( ) Variable)
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) holds 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) / (x : ( ( ) ( ) Variable) ,y : ( ( ) ( ) 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 ) ) c= ((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) } : ( ( ) ( finite ) set ) ) : ( ( ) ( ) Element of K27(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) \/ {y : ( ( ) ( ) Variable) } : ( ( ) ( finite ) set ) : ( ( ) ( ) set ) ;

theorem :: ZFMODEL2:2
for y, x being ( ( ) ( ) Variable)
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) st not y : ( ( ) ( ) 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 ) Element of K27(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
( ( 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 ) ) implies 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) / (x : ( ( ) ( ) Variable) ,y : ( ( ) ( ) 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 ) ) = ((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) } : ( ( ) ( finite ) set ) ) : ( ( ) ( ) Element of K27(VAR : ( ( ) ( non empty ) Element of K27(NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) \/ {y : ( ( ) ( ) Variable) } : ( ( ) ( finite ) set ) : ( ( ) ( ) set ) ) & ( 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 ) ) implies 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) / (x : ( ( ) ( ) Variable) ,y : ( ( ) ( ) 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 ) ) = 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 ) ) ) ) ;

registration
let H be ( ( 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) ;
cluster 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 ) M6( NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) -> finite ;
end;

theorem :: ZFMODEL2:3
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) holds
( ex i being ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) 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 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 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
j : ( ( ) ( ext-real V18() V19() V20() V21() V25() V26() ) Element of NAT : ( ( ) ( non empty V19() V20() V21() ) Element of K27(REAL : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) < i : ( ( ) ( ) Variable) & not for x being ( ( ) ( ) Variable) holds 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 ) ) ) ;

theorem :: ZFMODEL2:4
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 :: ZFMODEL2:5
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 :: ZFMODEL2:6
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 :: ZFMODEL2:7
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( 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 ) ) / (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 ) ) / (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 ) ) & (((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 ) ) / (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 :: ZFMODEL2:8
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 ) ) )) : ( ( 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 ) ) / (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) ,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 ) ) )) : ( ( 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 ) ) = (((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 :: ZFMODEL2:9
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 :: ZFMODEL2:10
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 :: ZFMODEL2:11
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 :: ZFMODEL2:12
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 :: ZFMODEL2:13
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 :: ZFMODEL2:14
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(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 ) ) |= 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 :: ZFMODEL2:15
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) 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 ) |= 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 ) |= 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 :: ZFMODEL2:16
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 :: ZFMODEL2:17
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 :: ZFMODEL2:18
for M being ( ( non empty ) ( non empty ) set )
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 ) holds
F : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) * G : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( Relation-like ) ( Relation-like Function-like ) set ) is_definable_in M : ( ( non empty ) ( non empty ) set ) ;

theorem :: ZFMODEL2:19
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 :: ZFMODEL2:20
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 :: ZFMODEL2:21
for M being ( ( non empty ) ( non empty ) set )
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_parametrically_definable_in M : ( ( non empty ) ( non empty ) set ) & G : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) is_parametrically_definable_in M : ( ( non empty ) ( non empty ) set ) holds
G : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) * F : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( Relation-like ) ( Relation-like Function-like ) set ) is_parametrically_definable_in M : ( ( non empty ) ( non empty ) set ) ;

theorem :: ZFMODEL2:22
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 ) ;

theorem :: ZFMODEL2:23
for M being ( ( non empty ) ( non empty ) set ) holds id M : ( ( non empty ) ( non empty ) set ) : ( ( V14(b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one 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 ) ) is_definable_in M : ( ( non empty ) ( non empty ) set ) ;

theorem :: ZFMODEL2:24
for M being ( ( non empty ) ( non empty ) set ) holds id M : ( ( non empty ) ( non empty ) set ) : ( ( V14(b1 : ( ( non empty ) ( non empty ) set ) ) ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined b1 : ( ( non empty ) ( non empty ) set ) -valued Function-like one-to-one 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 ) ) is_parametrically_definable_in M : ( ( non empty ) ( non empty ) set ) ;