
theorem Th23: :: Generalized YELLOW_2:19
  for S, T being reflexive antisymmetric non empty RelStr, f being
  Function of S, T st f is filtered-infs-preserving holds f is monotone
proof
  let S, T be reflexive antisymmetric non empty RelStr, f be Function of S, T;
  assume
A1: f is filtered-infs-preserving;
  let x, y be Element of S such that
A2: x <= y;
A3: dom f = the carrier of S by FUNCT_2:def 1;
A4: for b being Element of S st {x, y} is_>=_than b holds x >= b by YELLOW_0:8;
A5: x <= x;
  then
A6: {x, y} is_>=_than x by A2,YELLOW_0:8;
  then
A7: ex_inf_of {x, y},S by A4,YELLOW_0:31;
  for a, b being Element of S st a in {x, y} & b in {x, y} ex z being
  Element of S st z in {x, y} & a >= z & b >= z
  proof
    let a, b be Element of S such that
A8: a in {x, y} & b in {x, y};
    take x;
    thus x in {x, y} by TARSKI:def 2;
    thus thesis by A2,A5,A8,TARSKI:def 2;
  end;
  then {x, y} is filtered non empty;
  then
A9: f preserves_inf_of {x, y} by A1;
  x = inf {x, y} by A6,A4,YELLOW_0:31;
  then inf(f.:{x, y}) = f.x by A7,A9;
  then
A10: f.x = inf{f.x, f.y} by A3,FUNCT_1:60;
  f.:{x, y} = {f.x, f.y} by A3,FUNCT_1:60;
  then ex_inf_of {f.x, f.y}, T by A7,A9;
  then {f.x, f.y} is_>=_than f.x by A10,YELLOW_0:def 10;
  hence f.x <= f.y by YELLOW_0:8;
end;
