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
  for X0 being open non empty SubSpace of X for A,C being Subset of X, B
  being Subset of X0 st C = the carrier of X0 & A = B holds C is dense & B is
  everywhere_dense iff A is everywhere_dense
proof
  let X0 be open non empty SubSpace of X;
  let A,C be Subset of X, B be Subset of X0;
  assume
A1: C = the carrier of X0;
  assume
A2: A = B;
  C is open by A1,TSEP_1:def 1;
  hence thesis by A1,A2,Th62;
end;
