
theorem
  for X being set, L being non empty RelStr, S being non empty SubRelStr
of L for f,g being Function of X, the carrier of S for f9,g9 being Function of
  X, the carrier of L st f9 = f & g9 = g & f <= g holds f9 <= g9
proof
  let X be set, L be non empty RelStr, S be non empty SubRelStr of L;
  let f,g be Function of X, the carrier of S;
  let f9,g9 be Function of X, the carrier of L such that
A1: f9 = f and
A2: g9 = g and
A3: f <= g;
  let x be set;
  assume
A4: x in X;
  then reconsider a = f.x, b = g.x as Element of S by FUNCT_2:5;
  reconsider a9 = a, b9 = b as Element of L by YELLOW_0:58;
  take a9, b9;
  thus a9 = f9.x & b9 = g9.x by A1,A2;
  ex a,b being Element of S st a = f.x & b = g.x & a <= b by A3,A4;
  hence thesis by YELLOW_0:59;
end;
