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;

theorem Th38:
  A is everywhere_dense & A c= B implies B is everywhere_dense
proof
  assume A is everywhere_dense;
  then
A1: Cl Int A = [#]X;
  assume A c= B;
  then Int A c= Int B by TOPS_1:19;
  then Cl Int A c= Cl Int B by PRE_TOPC:19;
  then [#]X = Cl Int B by A1;
  hence thesis;
end;
