reserve X, Y for non empty set;

theorem Th3:
  for R,S being Membership_Func of X holds R = S iff R c= S & S c= R
proof
  let R,S be Membership_Func of X;
  thus R=S implies R c= S & S c= R;
  assume
A1: R c= S & S c= R;
  for x being Element of X holds R.x = S.x
  proof
    let x be Element of X;
    R.x <= S.x & S.x <= R.x by A1;
    hence thesis by XXREAL_0:1;
  end;
  hence thesis by Th1;
end;
