
theorem
  for S being non empty TopSpace, T being non empty TopSpace-like
TopRelStr, f, g being Function of S, T, x, y being Element of ContMaps(S,T) st
  x = f & y = g holds x <= y iff f <= g
proof
  let S be non empty TopSpace, T be non empty TopSpace-like TopRelStr, f, g be
  Function of S, T, x, y be Element of ContMaps(S,T) such that
A1: x = f & y = g;
A2: ContMaps(S,T) is full SubRelStr of T |^ the carrier of S by WAYBEL24:def 3;
  then reconsider x1 = x, y1 = y as Element of T |^ the carrier of S by
YELLOW_0:58;
  hereby
    assume x <= y;
    then x1 <= y1 by A2,YELLOW_0:59;
    hence f <= g by A1,WAYBEL10:11;
  end;
  assume f <= g;
  then x1 <= y1 by A1,WAYBEL10:11;
  hence thesis by A2,YELLOW_0:60;
end;
