
theorem Th25:
  for S,S9 being non empty RelStr for T,T9 being non empty
  reflexive antisymmetric RelStr st the RelStr of S = the RelStr of S9 & the
  RelStr of T = the RelStr of T9 holds UPS(S,T) = UPS(S9,T9)
proof
  let S,S9 be non empty RelStr;
  let T,T9 be non empty reflexive antisymmetric RelStr;
  assume that
A1: the RelStr of S = the RelStr of S9 and
A2: the RelStr of T = the RelStr of T9;
  T |^ the carrier of S = T9 |^ the carrier of S9 by A1,A2,Th15;
  then
A3: UPS(S9,T9) is full SubRelStr of T |^ the carrier of S by Def4;
A4: the carrier of UPS(S,T) = the carrier of UPS(S9,T9)
  proof
    thus the carrier of UPS(S,T) c= the carrier of UPS(S9,T9)
    proof
      let x be object;
      assume x in the carrier of UPS(S,T);
      then reconsider x1=x as directed-sups-preserving Function of S,T by Def4;
      reconsider y=x1 as Function of S9,T9 by A1,A2;
      y is directed-sups-preserving
      proof
        let X be Subset of S9;
        reconsider Y=X as Subset of S by A1;
        assume X is non empty directed;
        then Y is non empty directed by A1,WAYBEL_0:3;
        then x1 preserves_sup_of Y by WAYBEL_0:def 37;
        hence thesis by A1,A2,WAYBEL_0:65;
      end;
      hence thesis by Def4;
    end;
    let x be object;
    assume x in the carrier of UPS(S9,T9);
    then reconsider x1=x as directed-sups-preserving Function of S9,T9 by Def4;
    reconsider y=x1 as Function of S,T by A1,A2;
    y is directed-sups-preserving
    proof
      let X be Subset of S;
      reconsider Y=X as Subset of S9 by A1;
      assume X is non empty directed;
      then Y is non empty directed by A1,WAYBEL_0:3;
      then x1 preserves_sup_of Y by WAYBEL_0:def 37;
      hence thesis by A1,A2,WAYBEL_0:65;
    end;
    hence thesis by Def4;
  end;
  UPS(S,T) is full SubRelStr of T |^ the carrier of S by Def4;
  hence thesis by A3,A4,YELLOW_0:57;
end;
