:: TMAP_1 semantic presentation

begin

registration
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X1, X2 be ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
cluster X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) -> non empty strict TopSpace-like ;
end;

definition
let A, B be ( ( non empty ) ( non empty ) set ) ;
let A1, A2 be ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ;
let f1 be ( ( Function-like V21(A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ;
let f2 be ( ( Function-like V21(A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ;
assume f1 : ( ( Function-like V21(A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
func f1 union f2 -> ( ( Function-like V21(A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) means :: TMAP_1:def 1
( it : ( ( ) ( ) Element of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like ) ( Relation-like A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f1 : ( ( ) ( ) a_neighborhood of A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) & it : ( ( ) ( ) Element of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) -defined B : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(A1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) \/ A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of bool A : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( ) ( ) a_neighborhood of A2 : ( ( ) ( ) Element of the carrier of B : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: TMAP_1:1
for A, B being ( ( non empty ) ( non empty ) set )
for A1, A2 being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) st A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) misses A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) holds
for f1 being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f2 being ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) holds
( f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:2
for A, B being ( ( non empty ) ( non empty ) set )
for A1, A2 being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for g1 being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for g2 being ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) st g : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = g1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) & g : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = g2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) holds
g : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = g1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union g2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: TMAP_1:3
for A, B being ( ( non empty ) ( non empty ) set )
for A1, A2 being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f1 being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f2 being ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) st f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) holds
f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: TMAP_1:4
for A, B being ( ( non empty ) ( non empty ) set )
for A1, A2, A3, A12, A23 being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) st A12 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) = A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & A23 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) = A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
for f1 being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f2 being ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f3 being ( ( Function-like V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) st f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f3 : ( ( Function-like V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) & f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f3 : ( ( Function-like V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) holds
for f12 being ( ( Function-like V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A12 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f23 being ( ( Function-like V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A23 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) st f12 : ( ( Function-like V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) & f23 : ( ( Function-like V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f3 : ( ( Function-like V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) holds
f12 : ( ( Function-like V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f3 : ( ( Function-like V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b6 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b5 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f23 : ( ( Function-like V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b7 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ;

theorem :: TMAP_1:5
for A, B being ( ( non empty ) ( non empty ) set )
for A1, A2 being ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) )
for f1 being ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) )
for f2 being ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,B : ( ( non empty ) ( non empty ) set ) ) st f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) | (A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) /\ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like ) Element of bool [:b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) holds
( ( A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) implies f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) & ( f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) implies A1 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) & ( A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) implies f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) & ( f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) union f2 : ( ( Function-like V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) \/ b4 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) Element of bool b1 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) = f1 : ( ( Function-like V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) ( non empty Relation-like b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) -defined b2 : ( ( non empty ) ( non empty ) set ) -valued Function-like total V21(b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) ) Function of b3 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) ,b2 : ( ( non empty ) ( non empty ) set ) ) implies A2 : ( ( non empty ) ( non empty ) Subset of ( ( ) ( non empty ) set ) ) is ( ( ) ( ) Subset of ( ( ) ( non empty ) set ) ) ) ) ;

begin

theorem :: TMAP_1:6
for X being ( ( ) ( ) TopStruct )
for X0 being ( ( ) ( ) SubSpace of X : ( ( ) ( ) TopStruct ) ) holds TopStruct(# the carrier of X0 : ( ( ) ( ) SubSpace of b1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) , the topology of X0 : ( ( ) ( ) SubSpace of b1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) Element of bool (bool the carrier of b2 : ( ( ) ( ) SubSpace of b1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict ) TopStruct ) is ( ( strict ) ( strict ) SubSpace of X : ( ( ) ( ) TopStruct ) ) ;

theorem :: TMAP_1:7
for X being ( ( ) ( ) TopStruct )
for X1, X2 being ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) st X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) = TopStruct(# the carrier of X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) , the topology of X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b3 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict TopSpace-like ) TopStruct ) holds
( X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( ) ( ) SubSpace of X : ( ( ) ( ) TopStruct ) ) iff X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( ) ( ) SubSpace of X : ( ( ) ( ) TopStruct ) ) ) ;

theorem :: TMAP_1:8
for X, X1, X2 being ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) st X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) = TopStruct(# the carrier of X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) , the topology of X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict TopSpace-like ) TopStruct ) holds
( X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( closed ) ( TopSpace-like closed ) SubSpace of X : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) ) iff X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( closed ) ( TopSpace-like closed ) SubSpace of X : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:9
for X, X1, X2 being ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) st X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) = TopStruct(# the carrier of X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) , the topology of X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( strict TopSpace-like ) TopStruct ) holds
( X1 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) ) iff X2 : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( TopSpace-like ) ( TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:10
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ex x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:11
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ex x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) or ex x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:12
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( ex x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & ex x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:13
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for F1 being ( ( ) ( ) Subset of )
for F2 being ( ( ) ( ) Subset of ) st F1 : ( ( ) ( ) Subset of ) is closed & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in F1 : ( ( ) ( ) Subset of ) & F2 : ( ( ) ( ) Subset of ) is closed & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in F2 : ( ( ) ( ) Subset of ) holds
ex H being ( ( ) ( ) Subset of ) st
( H : ( ( ) ( ) Subset of ) is closed & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in H : ( ( ) ( ) Subset of ) & H : ( ( ) ( ) Subset of ) c= F1 : ( ( ) ( ) Subset of ) \/ F2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) set ) ) ;

theorem :: TMAP_1:14
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for U1 being ( ( ) ( ) Subset of )
for U2 being ( ( ) ( ) Subset of ) st U1 : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in U1 : ( ( ) ( ) Subset of ) & U2 : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in U2 : ( ( ) ( ) Subset of ) holds
ex V being ( ( ) ( ) Subset of ) st
( V : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in V : ( ( ) ( ) Subset of ) & V : ( ( ) ( ) Subset of ) c= U1 : ( ( ) ( ) Subset of ) \/ U2 : ( ( ) ( ) Subset of ) : ( ( ) ( ) set ) ) ;

theorem :: TMAP_1:15
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
for A1 being ( ( ) ( ) a_neighborhood of x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for A2 being ( ( ) ( ) a_neighborhood of x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ex V being ( ( ) ( ) Subset of ) st
( V : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in V : ( ( ) ( ) Subset of ) & V : ( ( ) ( ) Subset of ) c= A1 : ( ( ) ( ) a_neighborhood of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) \/ A2 : ( ( ) ( ) a_neighborhood of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ) ;

theorem :: TMAP_1:16
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
for A1 being ( ( ) ( ) a_neighborhood of x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for A2 being ( ( ) ( ) a_neighborhood of x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ex A being ( ( ) ( ) a_neighborhood of x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st A : ( ( ) ( ) a_neighborhood of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) c= A1 : ( ( ) ( ) a_neighborhood of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) \/ A2 : ( ( ) ( ) a_neighborhood of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) set ) ;

theorem :: TMAP_1:17
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:18
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:19
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:20
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = TopStruct(# the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) ;

theorem :: TMAP_1:21
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = TopStruct(# the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) ;

theorem :: TMAP_1:22
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for Y1, X1, Y2, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:23
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for Y1, Y2, X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:24
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:25
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:26
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies ( (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies ( (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ) ;

theorem :: TMAP_1:27
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ) ;

theorem :: TMAP_1:28
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = TopStruct(# the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = TopStruct(# the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) ) ;

theorem :: TMAP_1:29
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ) ;

theorem :: TMAP_1:30
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, Y1, X2, Y2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:31
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, Y1, Y2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:32
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, Y1, Y2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = (Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:33
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, Y1, Y2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is not ( ( ) ( TopSpace-like ) SubSpace of Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = (Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
( Y1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union Y2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:34
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( not X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( not X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ;

theorem :: TMAP_1:35
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ;

theorem :: TMAP_1:36
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) implies ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ) ;

theorem :: TMAP_1:37
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( ( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & ( ( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ;

theorem :: TMAP_1:38
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for X0 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
X0 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( closed ) ( TopSpace-like closed ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:39
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for X0 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
X0 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;

theorem :: TMAP_1:40
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for X0 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X0 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( closed ) ( TopSpace-like closed ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( closed ) ( TopSpace-like closed ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ;

theorem :: TMAP_1:41
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for X0 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X0 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ;

begin

definition
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
pred f is_continuous_at x means :: TMAP_1:def 2
 for G being ( ( ) ( ) a_neighborhood of f : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) . x : ( ( ) ( ) Element of the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ex H being ( ( ) ( ) a_neighborhood of x : ( ( ) ( ) Element of the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) st f : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) .: H : ( ( ) ( ) a_neighborhood of x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) : ( ( ) ( non empty ) set ) ) c= G : ( ( ) ( ) a_neighborhood of f : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ;
end;

notation
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
antonym f is_not_continuous_at x for f is_continuous_at x;
end;

theorem :: TMAP_1:42
errorfrm ;

theorem :: TMAP_1:43
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff for G being ( ( ) ( ) Subset of ) st G : ( ( ) ( ) Subset of ) is open & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) in G : ( ( ) ( ) Subset of ) holds
ex H being ( ( ) ( ) Subset of ) st
( H : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in H : ( ( ) ( ) Subset of ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: H : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= G : ( ( ) ( ) Subset of ) ) ) ;

theorem :: TMAP_1:44
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous iff for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:45
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) st the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of Z : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & the topology of Z : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the topology of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:46
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) st the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & the topology of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:47
for Z, X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) & f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at y : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:48
errorfrm ;

theorem :: TMAP_1:49
for Y, Z, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous holds
g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:50
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:51
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) st the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of Z : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & the topology of Z : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the topology of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:52
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) st the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & the topology of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) holds
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

definition
let S, T be ( ( ) ( ) 1-sorted ) ;
let f be ( ( Function-like V21( the carrier of S : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of S : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -defined the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) -valued Function-like V21( the carrier of S : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) , the carrier of T : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) ;
let R be ( ( ) ( ) 1-sorted ) ;
assume the carrier of R : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) c= the carrier of S : ( ( ) ( ) 1-sorted ) : ( ( ) ( ) set ) ;
func f | R -> ( ( Function-like V21( the carrier of R : ( ( ) ( ) Element of the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) , the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of R : ( ( ) ( ) Element of the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like V21( the carrier of R : ( ( ) ( ) Element of the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) , the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) equals :: TMAP_1:def 3
f : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | the carrier of R : ( ( ) ( ) Element of the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like ) Element of bool [: the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of T : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
redefine func f | X0 equals :: TMAP_1:def 4
f : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | the carrier of X0 : ( ( ) ( ) Element of the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) : ( ( Function-like ) ( Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: TMAP_1:53
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f0 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = f0 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f0 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:54
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st TopStruct(# the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) holds
f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:55
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:56
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for B being ( ( ) ( ) Subset of ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:57
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ex h being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st h : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:58
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:59
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for A being ( ( ) ( ) Subset of )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ) c= the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) & A : ( ( ) ( ) Subset of ) is ( ( ) ( ) a_neighborhood of x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:60
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for A being ( ( ) ( ) Subset of )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ) is open & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) & A : ( ( ) ( ) Subset of ) c= the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:61
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X0 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff f : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

registration
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let X0 be ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
cluster f : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V21( the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like V21( the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) continuous ;
end;

theorem :: TMAP_1:62
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds (g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * (f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:63
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous holds
(g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: TMAP_1:64
for X, Y, Z being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
(g : ( ( Function-like V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

definition
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0, X1 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
let g be ( ( Function-like V21( the carrier of X0 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of X0 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X0 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) ;
assume X1 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ;
func g | X1 -> ( ( Function-like V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) equals :: TMAP_1:def 5
g : ( ( ) ( ) a_neighborhood of X0 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) | the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) : ( ( Function-like ) ( Relation-like the carrier of X0 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like ) Element of bool [: the carrier of X0 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: TMAP_1:65
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) = (g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ;

theorem :: TMAP_1:66
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for g1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) = g1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:67
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:68
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) c= the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) .: A : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:69
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for B being ( ( ) ( ) Subset of ) st g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = (g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) " B : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:70
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:71
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
(f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:72
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds (g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:73
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f0 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) = X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f0 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f0 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:74
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:75
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:76
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for A being ( ( ) ( ) Subset of )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ) c= the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) & A : ( ( ) ( ) Subset of ) is ( ( ) ( ) a_neighborhood of x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:77
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for A being ( ( ) ( ) Subset of )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ) is open & x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) & A : ( ( ) ( ) Subset of ) c= the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:78
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for A being ( ( ) ( ) Subset of )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st A : ( ( ) ( ) Subset of ) is open & x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) & A : ( ( ) ( ) Subset of ) c= the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) & x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:79
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:80
for Y, X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:81
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st TopStruct(# the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the topology of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) Element of bool (bool the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:82
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
g : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:83
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X0, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) & X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) holds
for g being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

registration
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
cluster id X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) -> Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) continuous ;
end;

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
func incl X0 -> ( ( Function-like V21( the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -defined the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) equals :: TMAP_1:def 6
(id X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty ) set ) : ( ( Function-like V21( the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -defined the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( non empty ) set ) ) ;
end;

notation
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
synonym X0 incl X for incl X0;
end;

theorem :: TMAP_1:84
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
(incl X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) = x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:85
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f0 being ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = f0 : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) holds
incl X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f0 : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:86
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * (incl X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Element of bool [: the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:87
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds incl X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

begin

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let A be ( ( ) ( ) Subset of ) ;
func A -extension_of_the_topology_of X -> ( ( ) ( ) Subset-Family of ) equals :: TMAP_1:def 7
 { (H : ( ( ) ( ) Subset of ) \/ (G : ( ( ) ( ) Subset of ) /\ A : ( ( non empty ) ( non empty ) set ) ) : ( ( ) ( ) Element of bool the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) Element of bool the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) where H, G is ( ( ) ( ) Subset of ) : ( H : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) Element of bool (bool the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & G : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) Element of bool (bool the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) ) } ;
end;

theorem :: TMAP_1:88
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) c= A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ;

theorem :: TMAP_1:89
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds { (G : ( ( ) ( ) Subset of ) /\ A : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) where G is ( ( ) ( ) Subset of ) : G : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) } c= A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ;

theorem :: TMAP_1:90
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A, C, D being ( ( ) ( ) Subset of ) st C : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) & D : ( ( ) ( ) Subset of ) in { (G : ( ( ) ( ) Subset of ) /\ A : ( ( ) ( ) Subset of ) ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) where G is ( ( ) ( ) Subset of ) : G : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) } holds
( C : ( ( ) ( ) Subset of ) \/ D : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) & C : ( ( ) ( ) Subset of ) /\ D : ( ( ) ( ) Subset of ) : ( ( ) ( ) Element of bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ) ;

theorem :: TMAP_1:91
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds A : ( ( ) ( ) Subset of ) in A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ;

theorem :: TMAP_1:92
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds
( A : ( ( ) ( ) Subset of ) in the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) iff the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ) ;

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let A be ( ( ) ( ) Subset of ) ;
func X modified_with_respect_to A -> ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) equals :: TMAP_1:def 8
 TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ,(A : ( ( non empty ) ( non empty ) set ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( ) Subset-Family of ) #) : ( ( strict ) ( non empty strict ) TopStruct ) ;
end;

registration
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let A be ( ( ) ( ) Subset of ) ;
cluster X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to A : ( ( ) ( ) Element of bool the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) -> non empty strict TopSpace-like ;
end;

theorem :: TMAP_1:93
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds
( the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to A : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & the topology of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to A : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ) ;

theorem :: TMAP_1:94
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for B being ( ( ) ( ) Subset of ) st B : ( ( ) ( ) Subset of ) = A : ( ( ) ( ) Subset of ) holds
B : ( ( ) ( ) Subset of ) is open ;

theorem :: TMAP_1:95
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds
( A : ( ( ) ( ) Subset of ) is open iff TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to A : ( ( ) ( ) Subset of ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) ;

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let A be ( ( ) ( ) Subset of ) ;
func modid (X,A) -> ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to A : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to A : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to A : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) equals :: TMAP_1:def 9
 id the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) : ( ( total ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V22( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V23( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) ) V69() V71() V72() V76() ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;
end;

theorem :: TMAP_1:96
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st not x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) in A : ( ( ) ( ) Subset of ) holds
modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) ) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:97
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) misses A : ( ( ) ( ) Subset of ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds (modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:98
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) = A : ( ( ) ( ) Subset of ) holds
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds (modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:99
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) misses A : ( ( ) ( ) Subset of ) holds
(modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:100
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of )
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) = A : ( ( ) ( ) Subset of ) holds
(modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:101
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for A being ( ( ) ( ) Subset of ) holds
( A : ( ( ) ( ) Subset of ) is open iff modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,A : ( ( ) ( ) Subset of ) ) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
func X modified_with_respect_to X0 -> ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) means :: TMAP_1:def 10
 for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) = the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) holds
it : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to A : ( ( ) ( ) Subset of ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) ;
end;

registration
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
cluster X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to X0 : ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) -> non empty strict TopSpace-like ;
end;

theorem :: TMAP_1:102
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) = the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) & ( for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) = the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
the topology of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) = A : ( ( ) ( ) Subset of ) -extension_of_the_topology_of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( ) Subset-Family of ) ) ) ;

theorem :: TMAP_1:103
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for Y0 being ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) st the carrier of Y0 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) : ( ( ) ( ) set ) = the carrier of X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) holds
Y0 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) ;

theorem :: TMAP_1:104
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) iff TopStruct(# the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the topology of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) Element of bool (bool the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( non empty ) set ) ) #) : ( ( strict ) ( non empty strict TopSpace-like ) TopStruct ) = X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) ) ;

definition
let X be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X0 be ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
func modid (X,X0) -> ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to X0 : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to X0 : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to X0 : ( ( non empty ) ( non empty ) set ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) means :: TMAP_1:def 11
 for A being ( ( ) ( ) Subset of ) st A : ( ( ) ( ) Subset of ) = the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) holds
it : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of X0 : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) = modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,A : ( ( ) ( ) Subset of ) ) : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to b1 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to b1 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) modified_with_respect_to b1 : ( ( ) ( ) Subset of ) ) : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopSpace) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) ;
end;

theorem :: TMAP_1:105
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = id X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Element of bool [: the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) :] : ( ( ) ( non empty Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:106
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds (modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:107
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x0 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds (modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x0 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:108
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0, X1 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
(modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:109
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds (modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) )) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X0 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:110
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X0 being ( ( ) ( TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X0 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( open ) ( TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) iff modid (X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,X0 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) modified_with_respect_to b2 : ( ( ) ( TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;

begin

theorem :: TMAP_1:111
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:112
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:113
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is_continuous_at x2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:114
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
for g being ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:115
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:116
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:117
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:118
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:119
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:120
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:121
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:122
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) iff ( f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TMAP_1:123
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_separated iff ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & ( for Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for g being ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st g : ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & g : ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
g : ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: TMAP_1:124
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_separated iff ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & ( for Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: TMAP_1:125
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_separated iff ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & ( for Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ) ;

begin

definition
let X, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let X1, X2 be ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ;
let f1 be ( ( Function-like V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let f2 be ( ( Function-like V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
assume ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or f1 : ( ( Function-like V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;
func f1 union f2 -> ( ( Function-like V21( the carrier of (X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( non empty Relation-like the carrier of (X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of (X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) means :: TMAP_1:def 12
( it : ( ( ) ( ) Element of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) = f1 : ( ( ) ( ) a_neighborhood of X1 : ( ( Function-like V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) ( non empty Relation-like the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) continuous ) Element of bool [: the carrier of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( non empty ) set ) ) ) & it : ( ( ) ( ) Element of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( Function-like V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) ( Relation-like the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) -defined the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) -valued Function-like total V21( the carrier of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) : ( ( ) ( non empty ) set ) , the carrier of Y : ( ( non empty ) ( non empty ) set ) : ( ( ) ( ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) = f2 : ( ( ) ( ) a_neighborhood of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ) ) );
end;

theorem :: TMAP_1:126
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for g being ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = (g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union (g : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:127
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for g being ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = (g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union (g : ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:128
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( ( (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) iff f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ;

theorem :: TMAP_1:129
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
( ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) & ( f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) implies X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) implies f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) & ( f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) implies X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) is ( ( ) ( TopSpace-like ) SubSpace of X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) ) ) ;

theorem :: TMAP_1:130
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:131
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2, X3 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f3 being ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) & ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f3 : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) & ( X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) or f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f3 : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) holds
(f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f3 : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of ((b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of ((b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of ((b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union (f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f3 : ( ( Function-like V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union (b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b5 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:132
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:133
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:134
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:135
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meets X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | (X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) meet b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:136
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:137
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:138
for X being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_separated iff ( X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) misses X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & ( for Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f1 being ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds f1 : ( ( Function-like V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b4 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) ) ) ;

theorem :: TMAP_1:139
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty ) ( non empty TopSpace-like ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) holds
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & X1 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ,X2 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) are_weakly_separated holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty ) ( non empty TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:140
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty closed ) ( non empty TopSpace-like closed ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;

theorem :: TMAP_1:141
for X, Y being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for X1, X2 being ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) )
for f1 being ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for f2 being ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st X : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) = X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X1 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) & (f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) | X2 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) = f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds
f1 : ( ( Function-like V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( Relation-like the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) union f2 : ( ( Function-like V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) ( non empty Relation-like the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of (b3 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) union b4 : ( ( non empty open ) ( non empty TopSpace-like open ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) ) : ( ( non empty strict ) ( non empty strict TopSpace-like ) SubSpace of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( Function-like V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) ( non empty Relation-like the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like total V21( the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) , the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) ) continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;