:: TOPALG_4 semantic presentation

begin

theorem :: TOPALG_4:1
for G, H being ( ( non empty ) ( non empty ) multMagma )
for x being ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) Element of ( ( ) ( non empty ) set ) ) ex g being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ex h being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) Element of ( ( ) ( non empty ) set ) ) = <*g : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,h : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*b1 : ( ( non empty ) ( non empty ) multMagma ) ,b2 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) )) ;

definition
let G1, G2, H1, H2 be ( ( non empty ) ( non empty ) multMagma ) ;
let f be ( ( Function-like quasi_total ) ( Relation-like the carrier of G1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of H1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let g be ( ( Function-like quasi_total ) ( Relation-like the carrier of G2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of H2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
func Gr2Iso (f,g) -> ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*G1 : ( ( ) ( ) TopStruct ) ,G2 : ( ( ) ( ) M2( bool (bool G1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) -defined the carrier of (product <*H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,H2 : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) means :: TOPALG_4:def 1
 for x being ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) Element of ( ( ) ( ) set ) ) ex x1 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) ex x2 being ( ( ) ( ) Element of ( ( ) ( ) set ) ) st
( x : ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) Element of ( ( ) ( non empty ) set ) ) = <*x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*G1 : ( ( ) ( ) TopStruct ) ,G2 : ( ( ) ( ) M2( bool (bool G1 : ( ( ) ( ) TopStruct ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) )) & it : ( ( ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued ) M2( bool H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) . x : ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) Element of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M2( the carrier of (product <*H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,H2 : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) )) = <*(f : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) . x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) M2( the carrier of H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) )) ,(g : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of G1 : ( ( ) ( ) TopStruct ) : ( ( ) ( ) set ) -valued Function-like quasi_total continuous ) Path of H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,H2 : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) ) . x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) M2( the carrier of H2 : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*H1 : ( ( Function-like quasi_total ) ( Relation-like [:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) -defined G1 : ( ( ) ( ) TopStruct ) -valued Function-like quasi_total ) M2( bool [:[:G1 : ( ( ) ( ) TopStruct ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) ,G1 : ( ( ) ( ) TopStruct ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,H2 : ( ( ) ( ) M2(G1 : ( ( ) ( ) TopStruct ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) )) );
end;

theorem :: TOPALG_4:2
for G1, G2, H1, H2 being ( ( non empty ) ( non empty ) multMagma )
for f being ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for x1 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) )
for x2 being ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) holds (Gr2Iso (f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*b1 : ( ( non empty ) ( non empty ) multMagma ) ,b2 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*b3 : ( ( non empty ) ( non empty ) multMagma ) ,b4 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . <*x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ,x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*b1 : ( ( non empty ) ( non empty ) multMagma ) ,b2 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*b3 : ( ( non empty ) ( non empty ) multMagma ) ,b4 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) )) = <*(f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x1 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) M2( the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) )) ,(g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x2 : ( ( ) ( ) Element of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( ) M2( the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*b3 : ( ( non empty ) ( non empty ) multMagma ) ,b4 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) )) ;

registration
let G1, G2, H1, H2 be ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ;
let f be ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) ;
let g be ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) ;
cluster Gr2Iso (f : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) , the carrier of H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,g : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) , the carrier of H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total multiplicative ;
end;

theorem :: TOPALG_4:3
for G1, G2, H1, H2 being ( ( non empty ) ( non empty ) multMagma )
for f being ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one holds
Gr2Iso (f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*b1 : ( ( non empty ) ( non empty ) multMagma ) ,b2 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*b3 : ( ( non empty ) ( non empty ) multMagma ) ,b4 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is one-to-one ;

theorem :: TOPALG_4:4
for G1, G2, H1, H2 being ( ( non empty ) ( non empty ) multMagma )
for f being ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) )
for g being ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) st f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is onto & g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is onto holds
Gr2Iso (f : ( ( Function-like quasi_total ) ( Relation-like the carrier of b1 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ,g : ( ( Function-like quasi_total ) ( Relation-like the carrier of b2 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty ) ( non empty ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*b1 : ( ( non empty ) ( non empty ) multMagma ) ,b2 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*b3 : ( ( non empty ) ( non empty ) multMagma ) ,b4 : ( ( non empty ) ( non empty ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is onto ;

theorem :: TOPALG_4:5
for G1, G2, H1, H2 being ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group)
for f being ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of G1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,H1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) )
for g being ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of G2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,H2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) st f : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) is bijective & g : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) is bijective holds
Gr2Iso (f : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) ,g : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -defined the carrier of b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*b1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*b3 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) ,b4 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) Group) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict V138() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is bijective ;

theorem :: TOPALG_4:6
errorfrm ;

begin

theorem :: TOPALG_4:7
for f, g being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = dom g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
pr1 <:f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ,g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) :> : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;

theorem :: TOPALG_4:8
for f, g being ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) st dom f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) = dom g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) : ( ( ) ( ) set ) holds
pr2 <:f : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ,g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) :> : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = g : ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) ;

definition
let S, T, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
let g be ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
:: original: <:
redefine func <:f,g:> -> ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) ;
end;

definition
let S, T, Y be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let f be ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
:: original: pr1
redefine func pr1 f -> ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) ;
:: original: pr2
redefine func pr2 f -> ( ( Function-like quasi_total ) ( Relation-like the carrier of Y : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( ) set ) ) ;
end;

theorem :: TOPALG_4:9
for Y, S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like quasi_total 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) ,b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds pr1 f : ( ( Function-like quasi_total 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) ,b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( 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 quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: TOPALG_4:10
for Y, S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for f being ( ( Function-like quasi_total 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) ,b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) holds pr2 f : ( ( Function-like quasi_total 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) ,b3 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like quasi_total ) ( 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 quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is continuous ;

theorem :: TOPALG_4:11
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected holds
s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected ;

theorem :: TOPALG_4:12
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected holds
t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected ;

theorem :: TOPALG_4:13
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected holds
for L being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) holds pr1 L : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPALG_4:14
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected holds
for L being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) holds pr2 L : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPALG_4:15
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected & t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected holds
[s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected ;

theorem :: TOPALG_4:16
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected & t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) are_connected holds
for L1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for L2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) holds <:L1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,L2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) :> : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ;

definition
let S, T be ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ;
let s1, s2 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t1, t2 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let L1 be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;
let L2 be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;
:: original: <:
redefine func <:L1,L2:> -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t1 : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -defined the carrier of s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) , the carrier of s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ,[s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t2 : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) ;
end;

definition
let S, T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let s be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let L1 be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;
let L2 be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;
:: original: <:
redefine func <:L1,L2:> -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) ;
end;

registration
let S, T be ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ;
cluster [:S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopStruct ) ,T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopStruct ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) -> strict TopSpace-like pathwise_connected ;
end;

definition
let S, T be ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ;
let s1, s2 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t1, t2 be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let L be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ;
:: original: pr1
redefine func pr1 L -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) ;
:: original: pr2
redefine func pr2 L -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total continuous ) Path of t1 : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -defined the carrier of s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) , the carrier of s1 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,t2 : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -defined the carrier of s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) , the carrier of s2 : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ) ;
end;

definition
let S, T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let s be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let L be ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ;
:: original: pr1
redefine func pr1 L -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) ;
:: original: pr2
redefine func pr2 L -> ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Loop of t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) ;
end;

theorem :: TOPALG_4:17
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for H being ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) are_homotopic holds
pr1 H : ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of b7 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,b8 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPALG_4:18
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for H being ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) are_homotopic holds
pr2 H : ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of b7 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,b8 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ;

theorem :: TOPALG_4:19
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) are_homotopic holds
p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic ;

theorem :: TOPALG_4:20
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) are_homotopic holds
p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic ;

theorem :: TOPALG_4:21
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for x, y being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for f being ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) )
for g being ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of x : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,y : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & x : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & y : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic & x : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,y : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic holds
<:f : ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of b9 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,b10 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) ,g : ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of b11 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,b12 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ) :> : ( ( Function-like quasi_total ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is ( ( ) ( Relation-like the carrier of [:K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) ,K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Homotopy of l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) ;

theorem :: TOPALG_4:22
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1, l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p, q being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for x, y being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & x : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & y : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & p : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,q : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic & x : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ,y : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) are_homotopic holds
l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ,l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) are_homotopic ;

theorem :: TOPALG_4:23
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2, s3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2, t3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for p2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected & [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected & p1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) holds
pr1 (l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) + l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) + p2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPALG_4:24
for S, T being ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace)
for s1, s2, s3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2, t3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) holds pr1 (l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) + l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = (pr1 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) + (pr1 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPALG_4:25
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2, s3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2, t3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for p1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )
for p2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) st [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected & [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) are_connected & p1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) & p2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) holds
pr2 (l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) + l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty V126() V127() V128() ) set ) , ( ( ) ( non empty ) set ) ) = p1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) + p2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

theorem :: TOPALG_4:26
for S, T being ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace)
for s1, s2, s3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2, t3 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l1 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) )
for l2 being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[s3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) holds pr2 (l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) + l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) = (pr2 l1 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) + (pr2 l2 : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Path of [b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,[b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b7 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total continuous ) Path of b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b8 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) ;

definition
let S, T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let s be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
func FGPrIso (s,t) -> ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) ,[s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) )) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) -defined the carrier of (product <*(pi_1 (S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( strict ) ( strict ) multMagma ) ,(pi_1 (T : ( ( ) ( ) set ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( strict ) ( strict ) multMagma ) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( ) set ) , ( ( ) ( ) set ) ) means :: TOPALG_4:def 2
 for x being ( ( ) ( ) Point of ( ( ) ( ) set ) ) ex l being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) st
( x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = Class ((EqRel ([:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) ,[s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) )) : ( ( ) ( Relation-like Loops [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) -defined Loops [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) -valued ) M2( bool [:(Loops [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) set ) ,(Loops [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( ) M2( bool (Loops [s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,T : ( ( ) ( ) set ) :] : ( ( strict TopSpace-like ) ( strict TopSpace-like ) TopStruct ) : ( ( ) ( ) set ) )) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) & it : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -defined the carrier of s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) M2( bool [: the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) , the carrier of s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) M2( the carrier of (product <*(pi_1 (S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( strict ) ( strict ) multMagma ) ,(pi_1 (T : ( ( ) ( ) set ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( strict ) ( strict ) multMagma ) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ) : ( ( strict ) ( strict ) multMagma ) : ( ( ) ( ) set ) )) = <*(Class ((EqRel (S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) ,s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( ) ( Relation-like Loops s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( ) set ) -defined Loops s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( ) set ) -valued ) M2( bool [:(Loops s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) ,(Loops s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr1 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of S : ( ( non empty V218() V219() V220() V221() ) ( non empty V218() V219() V220() V221() ) L15()) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) )) : ( ( ) ( ) M2( bool (Loops s : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) ,(Class ((EqRel (T : ( ( ) ( ) set ) ,t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )) : ( ( ) ( Relation-like Loops t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( ) set ) -defined Loops t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( ) set ) -valued ) M2( bool [:(Loops t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) ,(Loops t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr2 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of T : ( ( ) ( ) set ) : ( ( ) ( ) set ) -valued Function-like quasi_total ) Loop of t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) )) : ( ( ) ( ) M2( bool (Loops t : ( ( non empty Group-like V96() ) ( non empty unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) : ( ( ) ( ) set ) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) );
end;

theorem :: TOPALG_4:27
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) = Class ((EqRel ([:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( ) ( non empty Relation-like Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty ) set ) -defined Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) ,(Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( ) M2( bool (Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) holds
(FGPrIso (s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() FinSequence-like FinSubsequence-like ) M2( the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) )) = <*(Class ((EqRel (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty Relation-like Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -defined Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr1 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( ) M2( bool (Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) ,(Class ((EqRel (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty Relation-like Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -defined Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr2 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( ) M2( bool (Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ;

theorem :: TOPALG_4:28
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for l being ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) holds (FGPrIso (s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) . (Class ((EqRel ([:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( ) ( non empty Relation-like Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty ) set ) -defined Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) ,(Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) )) : ( ( ) ( ) M2( bool (Loops [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) : ( ( ) ( ) set ) = <*(Class ((EqRel (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty Relation-like Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -defined Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr1 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( ) M2( bool (Loops b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) ,(Class ((EqRel (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( ) ( non empty Relation-like Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -defined Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( non empty ) set ) -valued total V197() V202() ) M2( bool [:(Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) ,(Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) ,(pr2 l : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of [b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) ) : ( ( ) ( Relation-like the carrier of K554() : ( ( ) ( non empty strict TopSpace-like V227() pathwise_connected ) SubSpace of K552() : ( ( TopSpace-like ) ( non empty strict TopSpace-like V227() ) TopStruct ) ) : ( ( ) ( non empty V126() V127() V128() ) set ) -defined the carrier of b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Loop of b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) )) : ( ( ) ( ) M2( bool (Loops b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( non empty ) set ) : ( ( ) ( ) set ) )) *> : ( ( ) ( non empty Relation-like K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like ) set ) ;

registration
let S, T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let s be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
cluster FGPrIso (s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like one-to-one quasi_total onto ;
end;

registration
let S, T be ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ;
let s be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
let t be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
cluster FGPrIso (s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ] : ( ( ) ( V24() ) M2( the carrier of [:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,s : ( ( ) ( ) M2( the carrier of S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) ,t : ( ( ) ( ) M2( the carrier of T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one quasi_total onto ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) -> Function-like quasi_total multiplicative ;
end;

theorem :: TOPALG_4:29
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds FGPrIso (s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one quasi_total onto multiplicative ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) is bijective ;

theorem :: TOPALG_4:30
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds pi_1 ([:S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , product <*(pi_1 (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,s : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,t : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) are_isomorphic ;

theorem :: TOPALG_4:31
for S, T being ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for f being ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (S : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) )
for g being ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (T : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) st f : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) is bijective & g : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) is bijective holds
(Gr2Iso (f : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) ,g : ( ( Function-like quasi_total multiplicative ) ( Relation-like the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Homomorphism of pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ) )) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like quasi_total multiplicative ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) * (FGPrIso (s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( Function-like quasi_total ) ( Relation-like the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like one-to-one quasi_total onto multiplicative ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -defined the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) -valued Function-like ) M2( bool [: the carrier of (pi_1 ([:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) ,[b3 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,b5 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like ) TopStruct ) : ( ( ) ( non empty ) set ) )) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) , the carrier of (product <*(pi_1 (b1 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b4 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (b2 : ( ( non empty TopSpace-like ) ( non empty TopSpace-like ) TopSpace) ,b6 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) : ( ( ) ( non empty ) set ) :] : ( ( ) ( Relation-like ) set ) : ( ( ) ( ) set ) )) is bijective ;

theorem :: TOPALG_4:32
for S, T being ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace)
for s1, s2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) )
for t1, t2 being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds pi_1 ([:S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) ,[s1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ,t1 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ] : ( ( ) ( V24() ) M2( the carrier of [:b1 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,b2 : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) :] : ( ( strict TopSpace-like ) ( non empty strict TopSpace-like pathwise_connected ) TopStruct ) : ( ( ) ( non empty ) set ) )) ) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) , product <*(pi_1 (S : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,s2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) ,(pi_1 (T : ( ( non empty TopSpace-like pathwise_connected ) ( non empty TopSpace-like pathwise_connected ) TopSpace) ,t2 : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) )) : ( ( strict ) ( non empty strict unital Group-like V96() V148() V149() V150() V151() V152() V153() ) multMagma ) *> : ( ( ) ( non empty Relation-like {1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) -defined K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) -defined Function-like total V32() V39(2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ) FinSequence-like FinSubsequence-like V173() V176() Group-like V178({1 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) ,2 : ( ( ) ( non empty V10() V11() V12() V30() V31() ext-real V126() V127() V128() V129() V130() V131() ) M3(K28() : ( ( ) ( V126() V127() V128() V132() ) set ) ,K32() : ( ( ) ( V126() V127() V128() V129() V130() V131() V132() ) M2( bool K28() : ( ( ) ( V126() V127() V128() V132() ) set ) : ( ( ) ( ) set ) )) )) } : ( ( ) ( V126() V127() V128() V129() V130() V131() ) set ) ) ) set ) : ( ( strict ) ( non empty strict unital Group-like V96() V138() V148() V149() V150() V151() V152() V153() ) multMagma ) are_isomorphic ;