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 Th54:
  for C, A being Subset of X, B being Subset of Y0 st C is closed
  & C c= the carrier of Y0 & A c= C & A = B holds Cl A = Cl B
proof
  let C, A be Subset of X, B be Subset of Y0;
  assume
A1: C is closed;
  assume
A2: C c= the carrier of Y0;
  assume A c= C;
  then Cl A c= Cl C by PRE_TOPC:19;
  then Cl A c= C by A1,PRE_TOPC:22;
  then
A3: Cl A = (Cl A) /\ [#]Y0 by A2,XBOOLE_1:1,28;
  assume A = B;
  hence thesis by A3,PRE_TOPC:17;
end;
