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;

theorem Th59:
  for A being Subset of X, B being Subset of X0 st A c= B holds A
  is dense implies B is dense
proof
  let A be Subset of X, B be Subset of X0;
A1: [#]X0 c= [#]X by PRE_TOPC:def 4;
  assume
A2: A c= B;
  then reconsider C = A as Subset of X0 by XBOOLE_1:1;
  assume A is dense;
  then Cl A = [#]X;
  then [#]X0 = (Cl A) /\ [#]X0 by A1,XBOOLE_1:28;
  then Cl C = [#]X0 by PRE_TOPC:17;
  then C is dense;
  hence thesis by A2,TOPS_1:44;
end;
