let Y be non empty TopStruct ; :: thesis: for A, B being non empty Subset of holds
( ( B is Subset of & A is Subset of ) iff TopStruct(# the carrier of (MaxADSspace A),the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B),the topology of (MaxADSspace B) #) )
let A, B be non empty Subset of ; :: thesis: ( ( B is Subset of & A is Subset of ) iff TopStruct(# the carrier of (MaxADSspace A),the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B),the topology of (MaxADSspace B) #) )
A1: the carrier of (MaxADSspace B) = MaxADSet B by Def18;
A2: the carrier of (MaxADSspace A) = MaxADSet A by Def18;
hence ( B is Subset of & A is Subset of implies TopStruct(# the carrier of (MaxADSspace A),the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B),the topology of (MaxADSspace B) #) ) by A1, Th37, TSEP_1:5; :: thesis: ( TopStruct(# the carrier of (MaxADSspace A),the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B),the topology of (MaxADSspace B) #) implies ( B is Subset of & A is Subset of ) )
assume TopStruct(# the carrier of (MaxADSspace A),the topology of (MaxADSspace A) #) = TopStruct(# the carrier of (MaxADSspace B),the topology of (MaxADSspace B) #) ; :: thesis: ( B is Subset of & A is Subset of )
hence ( B is Subset of & A is Subset of ) by A2, A1, Th34; :: thesis: verum