reserve X for TopStruct,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A for Subset of X;
reserve X for TopSpace,
  A,B for Subset of X;
reserve X for non empty TopSpace,
  A, B for Subset of X;
reserve D for Subset of X;
reserve Y0 for SubSpace of X;
reserve X0 for SubSpace of X;
reserve X0 for non empty SubSpace of X;

theorem Th62:
  for C, A being Subset of X, B being Subset of X0 st C is open &
  C c= the carrier of X0 & A c= C & A = B holds C is dense & B is
  everywhere_dense iff A is everywhere_dense
proof
  let C, A be Subset of X, B be Subset of X0;
  assume
A1: C is open;
  assume C c= the carrier of X0;
  then reconsider E = C as Subset of X0;
A2: E is open by A1,TOPS_2:25;
  assume
A3: A c= C;
  assume
A4: A = B;
A5: Int B c= B by TOPS_1:16;
  then reconsider D = Int B as Subset of X by A4,XBOOLE_1:1;
  Int B c= Int E by A3,A4,TOPS_1:19;
  then
A6: Int B c= E by A2,TOPS_1:23;
  then
A7: D is open by A1,TSEP_1:9;
  thus C is dense & B is everywhere_dense implies A is everywhere_dense
  proof
    assume
A8: C is dense;
    assume B is everywhere_dense;
    then Int B is dense;
    then D is dense by A6,A8,Th60;
    then Int A is dense by A4,A5,A7,TOPS_1:24,44;
    hence thesis;
  end;
  thus A is everywhere_dense implies C is dense & B is everywhere_dense
  by A3,Th33,Th38,A4,Th61;
end;
