let X1, X2 be TopSpace; :: thesis: for D1 being Subset of X1
for D2 being Subset of X2 st D1 c= D2 & TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) holds
Int D1 c= Int D2
let D1 be Subset of X1; :: thesis: for D2 being Subset of X2 st D1 c= D2 & TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) holds
Int D1 c= Int D2
let D2 be Subset of X2; :: thesis: ( D1 c= D2 & TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) implies Int D1 c= Int D2 )
assume A1: D1 c= D2 ; :: thesis: ( not TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) or Int D1 c= Int D2 )
then reconsider C2 = D1 as Subset of X2 by XBOOLE_1:1;
assume TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) ; :: thesis: Int D1 c= Int D2
then ( Int D1 = Int C2 & Int C2 c= Int D2 ) by A1, Th77, TOPS_1:48;
hence Int D1 c= Int D2 ; :: thesis: verum