
theorem Th19:
  for X,Y being non empty set, T being non empty Poset for S1
  being full non empty SubRelStr of (T|^X)|^Y for S2 being full non empty
SubRelStr of (T|^Y)|^X for F being Function of S1, S2 st F is commuting holds F
  is monotone
proof
  let X,Y be non empty set, T be non empty Poset;
  let S1 be full non empty SubRelStr of (T|^X)|^Y;
  let S2 be full non empty SubRelStr of (T|^Y)|^X;
  let F be Function of S1, S2 such that
  for x being set st x in dom F holds x is Function-yielding Function and
A1: for f being Function st f in dom F holds F.f = commute f;
  let f,g be Element of S1;
A2: dom F = the carrier of S1 by FUNCT_2:def 1;
  then
A3: F.g = commute g by A1;
  reconsider Fa = F.f, Fb = F.g as Element of (T|^Y)|^X by YELLOW_0:58;
  reconsider a = f, b = g as Element of (T|^X)|^Y by YELLOW_0:58;
A4: the carrier of (T|^X)|^Y = Funcs(Y, the carrier of T|^X) by YELLOW_1:28
    .= Funcs(Y, Funcs(X, the carrier of T)) by YELLOW_1:28;
  assume f <= g;
  then
A5: a <= b by YELLOW_0:59;
A6: F.f = commute a by A2,A1;
  now
    let x be Element of X;
    now
      let y be Element of Y;
A7:   Fa.x.y = a.y.x by A4,A6,FUNCT_6:56;
A8:   Fb.x.y = b.y.x by A4,A3,FUNCT_6:56;
      a.y <= b.y by A5,Th14;
      hence Fa.x.y <= Fb.x.y by A7,A8,Th14;
    end;
    hence Fa.x <= Fb.x by Th14;
  end;
  then Fa <= Fb by Th14;
  hence thesis by YELLOW_0:60;
end;
