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 Th56:
  for A being Subset of X, B being Subset of Y0 st A c= B holds Int A c= Int B
proof
  let A be Subset of X, B be Subset of Y0;
A1: Int A c= A by TOPS_1:16;
  assume A c= B;
  then
A2: Int A c= B by A1,XBOOLE_1:1;
  then reconsider C = Int A as Subset of Y0 by XBOOLE_1:1;
  C is open by TOPS_2:25;
  hence thesis by A2,TOPS_1:24;
end;
