reserve FT for non empty RelStr;
reserve A for Subset of FT;
reserve T for non empty TopStruct;
reserve FMT for non empty FMT_Space_Str;
reserve x, y for Element of FMT;
reserve A, B, W, V for Subset of FMT;

theorem Th24:
  for FMT being non empty FMT_Space_Str, A, B being Subset of FMT
  holds A c= B implies A^Fob c= B^Fob
proof
  let FMT be non empty FMT_Space_Str;
  let A, B be Subset of FMT;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in A^Fob;
  then reconsider y=x as Element of FMT;
  for W being Subset of FMT st W in U_FMT y holds W meets B
  proof
    let W be Subset of FMT;
    assume W in U_FMT y;
    then W meets A by A2,Th20;
    then ex z being object st z in W & z in A by XBOOLE_0:3;
    hence W /\ B <> {} by A1,XBOOLE_0:def 4;
  end;
  hence thesis;
end;
