let X1, X2 be non empty 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 everywhere_dense holds
D2 is everywhere_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 everywhere_dense holds
D2 is everywhere_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 everywhere_dense implies D2 is everywhere_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 everywhere_dense or D2 is everywhere_dense )
assume A2: TopStruct(# the carrier of X1,the topology of X1 #) = TopStruct(# the carrier of X2,the topology of X2 #) ; :: thesis: ( not D1 is everywhere_dense or D2 is everywhere_dense )
assume D1 is everywhere_dense ; :: thesis: D2 is everywhere_dense
then ( Int D1 is dense & Int D1 c= Int D2 ) by A1, A2, Th35, Th78;
then Int D2 is dense by A2, Th83;
hence D2 is everywhere_dense by Th35; :: thesis: verum