:: REAL_LAT semantic presentation

begin

definition
func minreal -> ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) means :: REAL_LAT:def 1
 for x, y being ( ( ) ( V11() ext-real V15() ) Real) holds it : ( ( ) ( ) LattStr ) . (x : ( ( ) ( V11() ext-real V15() ) Real) ,y : ( ( ) ( V11() ext-real V15() ) Real) ) : ( ( ) ( V11() ext-real V15() ) Element of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) = min (x : ( ( ) ( V11() ext-real V15() ) Real) ,y : ( ( ) ( V11() ext-real V15() ) Real) ) : ( ( ) ( V11() ext-real V15() ) Element of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ;
func maxreal -> ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) means :: REAL_LAT:def 2
 for x, y being ( ( ) ( V11() ext-real V15() ) Real) holds it : ( ( ) ( ) LattStr ) . (x : ( ( ) ( V11() ext-real V15() ) Real) ,y : ( ( ) ( V11() ext-real V15() ) Real) ) : ( ( ) ( V11() ext-real V15() ) Element of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) = max (x : ( ( ) ( V11() ext-real V15() ) Real) ,y : ( ( ) ( V11() ext-real V15() ) Real) ) : ( ( ) ( V11() ext-real V15() ) Element of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ;
end;

definition
func Real_Lattice -> ( ( strict ) ( strict ) LattStr ) equals :: REAL_LAT:def 3
 LattStr(# REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) ;
end;

registration
cluster the carrier of Real_Lattice : ( ( strict ) ( strict ) LattStr ) : ( ( ) ( ) set ) -> real-membered ;
end;

registration
cluster Real_Lattice : ( ( strict ) ( strict ) LattStr ) -> non empty strict ;
end;

registration
let a, b be ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ;
identify ;
identify ;
end;

registration
cluster Real_Lattice : ( ( strict ) ( non empty strict ) LattStr ) -> strict Lattice-like ;
end;

theorem :: REAL_LAT:1
for p, q being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: REAL_LAT:2
for p, q being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: REAL_LAT:3
for p, q, r being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds
( maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_LAT:4
for p, q, r being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds
( minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_LAT:5
for p, q being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds
( maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) ;

theorem :: REAL_LAT:6
for q, p being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds
( minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) & minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ) ;

theorem :: REAL_LAT:7
for q, p, r being ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) holds minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,(maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . ((minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,p : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ,(minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) . (q : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) ,r : ( ( ) ( V11() ext-real V15() ) Element of ( ( ) ( complex-membered ext-real-membered real-membered ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ;

registration
cluster Real_Lattice : ( ( strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -> strict distributive ;
end;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func maxfuncreal A -> ( ( V21() quasi_total ) ( Relation-like K7((Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) means :: REAL_LAT:def 4
 for f, g being ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds it : ( ( ) ( ) set ) . (f : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = maxreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) .: (f : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;
func minfuncreal A -> ( ( V21() quasi_total ) ( Relation-like K7((Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) means :: REAL_LAT:def 5
 for f, g being ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds it : ( ( ) ( ) set ) . (f : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = minreal : ( ( V21() quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) BinOp of REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) .: (f : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like A : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;
end;

theorem :: REAL_LAT:8
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:9
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:10
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:11
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:12
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:13
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:14
for A being ( ( non empty ) ( non empty ) set )
for g, f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:15
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:16
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:17
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:18
for A being ( ( non empty ) ( non empty ) set )
for g, f being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:19
for A being ( ( non empty ) ( non empty ) set )
for f, g being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

theorem :: REAL_LAT:20
for A being ( ( non empty ) ( non empty ) set )
for f, g, h being ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (A : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,g : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (f : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,h : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) )) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ) : ( ( ) ( Relation-like b1 : ( ( non empty ) ( non empty ) set ) -defined REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) -valued V21() quasi_total ) Element of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ;

definition
let A be ( ( non empty ) ( non empty ) set ) ;
func RealFunc_Lattice A -> ( ( non empty strict ) ( non empty strict ) LattStr ) equals :: REAL_LAT:def 6
 LattStr(# (Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(maxfuncreal A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) ,(minfuncreal A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of A : ( ( non empty Lattice-like ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) #) : ( ( strict ) ( non empty strict ) LattStr ) ;
end;

registration
let A be ( ( non empty ) ( non empty ) set ) ;
cluster RealFunc_Lattice A : ( ( non empty ) ( non empty ) set ) : ( ( non empty strict ) ( non empty strict ) LattStr ) -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing for ( ( non empty ) ( non empty ) LattStr ) ;
end;

theorem :: REAL_LAT:21
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: REAL_LAT:22
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: REAL_LAT:23
for A being ( ( non empty ) ( non empty ) set )
for p, q, r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_LAT:24
for A being ( ( non empty ) ( non empty ) set )
for p, q, r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: REAL_LAT:25
for A being ( ( non empty ) ( non empty ) set )
for p, q being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) ;

theorem :: REAL_LAT:26
for A being ( ( non empty ) ( non empty ) set )
for q, p being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds
( (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) & (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ) ;

theorem :: REAL_LAT:27
for A being ( ( non empty ) ( non empty ) set )
for q, p, r being ( ( ) ( ) Element of ( ( ) ( ) set ) ) holds (minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,((maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) = (maxfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,p : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ,((minfuncreal A : ( ( non empty ) ( non empty ) set ) ) : ( ( V21() quasi_total ) ( Relation-like K7((Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ,(Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) )) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) : ( ( ) ( ) set ) -defined Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) -valued V21() quasi_total ) BinOp of Funcs (b1 : ( ( non empty ) ( non empty ) set ) ,REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) : ( ( ) ( non empty ) FUNCTION_DOMAIN of b1 : ( ( non empty ) ( non empty ) set ) , REAL : ( ( ) ( non empty V33() complex-membered ext-real-membered real-membered V138() ) set ) ) ) . (q : ( ( ) ( ) Element of ( ( ) ( ) set ) ) ,r : ( ( ) ( ) Element of ( ( ) ( ) set ) ) )) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ;

theorem :: REAL_LAT:28
for A being ( ( non empty ) ( non empty ) set ) holds RealFunc_Lattice A : ( ( non empty ) ( non empty ) set ) : ( ( non empty strict ) ( non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like ) LattStr ) is ( ( non empty Lattice-like distributive ) ( non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular ) D_Lattice) ;