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;
reserve X1,X2 for TopStruct;
reserve X1,X2 for TopSpace;
reserve D1 for Subset of X1,
  D2 for Subset of X2;
reserve X1,X2 for non empty TopSpace;
reserve D1 for Subset of X1,
  D2 for Subset of X2;

theorem
  D1 c= D2 & the TopStruct of X1 = the TopStruct of X2 implies (D1 is
  everywhere_dense implies D2 is everywhere_dense)
proof
  assume
A1: D1 c= D2;
  assume
A2: the TopStruct of X1 = the TopStruct of X2;
  assume D1 is everywhere_dense;
  then Int D1 is dense;
  then Int D2 is dense by A1,A2,Th78,Th83;
  hence thesis;
end;
