reserve FT for non empty RelStr;
reserve x, y, z for Element of FT;
reserve A for Subset of FT;
reserve F for Subset of FT;

theorem Th14:
  for FT being non empty RelStr, A, B being Subset of FT holds A
  c= B implies A^b c= B^b
proof
  let FT be non empty RelStr;
  let A, B be Subset of FT;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in A^b;
  then reconsider y=x as Element of FT;
  U_FT y meets A by A2,Th8;
  then ex w being object st w in U_FT y & w in A by XBOOLE_0:3;
  then U_FT y meets B by A1,XBOOLE_0:3;
  hence thesis;
end;
