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;

theorem Th53:
  for A being Subset of X, B being Subset of Y0 st B c= A holds Cl B c= Cl A
proof
  let A be Subset of X, B be Subset of Y0;
  assume
A1: B c= A;
  then reconsider D = B as Subset of X by XBOOLE_1:1;
  Cl B = (Cl D) /\ [#]Y0 by PRE_TOPC:17;
  then
A2: Cl B c= Cl D by XBOOLE_1:17;
  Cl D c= Cl A by A1,PRE_TOPC:19;
  hence thesis by A2,XBOOLE_1:1;
end;
