
theorem
  for X being set, L being non empty RelStr for S being full 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 & f9 <= g9 holds f <= g
proof
  let X be set, L be non empty RelStr;
  let S be full 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: f9 <= g9;
  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;
  take a, b;
  thus a = f.x & b = g.x;
  ex a9,b9 being Element of L st a9 = a & b9 = b & a9 <= b9 by A1,A2,A3,A4;
  hence thesis by YELLOW_0:60;
end;
