reserve x, X, Y for set;

theorem
  for L, M being non empty RelStr, f,g being Function of L, M holds f <=
  g iff for x being Element of L holds f.x <= g.x
proof
  let L, M be non empty RelStr, f,g be Function of L, M;
  hereby
    assume
A1: f <= g;
    let x be Element of L;
    ex m1,m2 being Element of M st m1 = f.x & m2 = g.x & m1 <= m2 by A1;
    hence f.x <= g.x;
  end;
  assume
A2: for x being Element of L holds f.x <= g.x;
  let x be set;
  assume x in the carrier of L;
  then reconsider x as Element of L;
  take f.x, g.x;
  thus thesis by A2;
end;
