reserve X, Y for non empty set;

theorem Th5:
  for R,S,T being Membership_Func of X holds R c= S & S c= T implies R c= T
proof
  let R,S,T be Membership_Func of X;
  assume
A1: R c= S & S c= T;
  for x being Element of X holds R.x <= T.x
  proof
    let x be Element of X;
    R.x <= S.x & S.x <= T.x by A1;
    hence thesis by XXREAL_0:2;
  end;
  hence thesis;
end;
