
theorem Th29:
  for S,T being non empty reflexive antisymmetric RelStr holds UPS
  (id S, id T) = id UPS(S, T)
proof
  let S,T be non empty reflexive antisymmetric RelStr;
A1: for x being object st x in the carrier of UPS(S, T)
  holds UPS(id S, id T).x = x
  proof
    let x be object;
    assume x in the carrier of UPS(S, T);
    then reconsider f=x as directed-sups-preserving Function of S, T by Def4;
    UPS(id S, id T).f = (id T)*f*(id S) by Def5
      .= f*(id S) by FUNCT_2:17;
    hence thesis by FUNCT_2:17;
  end;
  dom UPS(id S, id T) = the carrier of UPS(S, T) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:17;
end;
