
theorem Th35:
  for S1, S2, T1, T2 being complete LATTICE for f being Function
  of S1, S2, g being Function of T1, T2 st f is isomorphic & g is isomorphic
  holds UPS(f, g) is isomorphic
proof
  let S1, S2, T1, T2 be complete LATTICE;
  let f be Function of S1, S2, g be Function of T1, T2;
  assume that
A1: f is isomorphic and
A2: g is isomorphic;
A3: g is sups-preserving Function of T1, T2 by A2,WAYBEL13:20;
A4: f is sups-preserving Function of S1, S2 by A1,WAYBEL13:20;
  then
A5: UPS(f,g) is directed-sups-preserving Function of UPS(S2, T1), UPS(S1, T2
  ) by A3,Th30;
  consider g9 being monotone Function of T2,T1 such that
A6: g*g9 = id T2 and
A7: g9*g = id T1 by A2,YELLOW16:15;
  g9 is isomorphic by A2,A6,A7,YELLOW16:15;
  then
A8: g9 is sups-preserving Function of T2, T1 by WAYBEL13:20;
  consider f9 being monotone Function of S2,S1 such that
A9: f*f9 = id S2 and
A10: f9*f = id S1 by A1,YELLOW16:15;
  f9 is isomorphic by A1,A9,A10,YELLOW16:15;
  then
A11: f9 is sups-preserving Function of S2, S1 by WAYBEL13:20;
  then
A12: UPS(f9,g9) is directed-sups-preserving Function of UPS(S1, T2), UPS( S2
  , T1 ) by A8,Th30;
A13: UPS(f9,g9)*UPS(f,g) = UPS(id S2, id T1) by A4,A3,A9,A7,A11,A8,Th28
    .= id UPS(S2,T1) by Th29;
  UPS(f,g)*UPS(f9,g9) = UPS(id S1, id T2) by A4,A3,A10,A6,A11,A8,Th28
    .= id UPS(S1,T2) by Th29;
  hence thesis by A13,A5,A12,YELLOW16:15;
end;
