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 #) & D1 is dense holds
D2 is dense
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 #) & D1 is dense holds
D2 is dense
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 #) & D1 is dense implies D2 is dense )
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 not D1 is dense or D2 is dense )
then reconsider C2 = D1 as Subset of X2 by XBOOLE_1:1;
assume A2: TopStruct(# the carrier of X1, the topology of X1 #) = TopStruct(# the carrier of X2, the topology of X2 #) ; :: thesis: ( not D1 is dense or D2 is dense )
assume D1 is dense ; :: thesis: D2 is dense
then A3: Cl D1 = the carrier of X1 ;
Cl D1 = Cl C2 by A2, Th80;
then C2 is dense by A2, A3;
hence D2 is dense by A1, TOPS_1:44; :: thesis: verum