
theorem Th28:
  for S1,S2,S3, T1,T2,T3 being non empty Poset for f1 being
  directed-sups-preserving Function of S2, S3 for f2 being
  directed-sups-preserving Function of S1, S2 for g1 being
  directed-sups-preserving Function of T1, T2 for g2 being
  directed-sups-preserving Function of T2, T3 holds UPS(f2, g2) * UPS(f1, g1) =
  UPS(f1*f2, g2*g1)
proof
  let S1,S2,S3,T1,T2,T3 be non empty Poset;
  let f1 be directed-sups-preserving Function of S2, S3;
  let f2 be directed-sups-preserving Function of S1, S2;
  let g1 be directed-sups-preserving Function of T1, T2;
  let g2 be directed-sups-preserving Function of T2, T3;
  reconsider F = f1*f2 as directed-sups-preserving Function of S1, S3 by
WAYBEL20:28;
  reconsider G = g2*g1 as directed-sups-preserving Function of T1, T3 by
WAYBEL20:28;
  for h being directed-sups-preserving Function of S3, T1 holds (UPS(f2,
  g2)*UPS(f1, g1)).h = G*h*F
  proof
    let h be directed-sups-preserving Function of S3, T1;
    g1*h is directed-sups-preserving Function of S3,T2 by WAYBEL20:28;
    then reconsider
    ghf=g1*h*f1 as directed-sups-preserving Function of S2, T2 by WAYBEL20:28;
    dom UPS(f1, g1) = the carrier of UPS(S3, T1) by FUNCT_2:def 1;
    then h in dom UPS(f1, g1) by Def4;
    then (UPS(f2, g2)*UPS(f1, g1)).h = UPS(f2, g2).(UPS(f1, g1).h) by
FUNCT_1:13
      .= UPS(f2, g2).(g1*h*f1) by Def5;
    hence (UPS(f2, g2)*UPS(f1, g1)).h = g2*(ghf)*f2 by Def5
      .= g2*((g1*h*f1)*f2) by RELAT_1:36
      .= g2*((g1*(h*f1))*f2) by RELAT_1:36
      .= g2*(g1*((h*f1)*f2)) by RELAT_1:36
      .= g2*(g1*(h*(f1*f2))) by RELAT_1:36
      .= g2*((g1*h)*(f1*f2)) by RELAT_1:36
      .= (g2*(g1*h))*(f1*f2) by RELAT_1:36
      .= G*h*F by RELAT_1:36;
  end;
  hence thesis by Def5;
end;
