
theorem Th43:
  for L being up-complete antisymmetric non empty reflexive
  RelStr st inf_op L is directed-sups-preserving holds for D1, D2 being non
  empty directed Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2)
proof
  let L be up-complete antisymmetric non empty reflexive RelStr such that
A1: inf_op L is directed-sups-preserving;
  let D1, D2 be non empty directed Subset of L;
  set X = [:D1,D2:], f = inf_op L;
A2: f preserves_sup_of X by A1;
A3: ex_sup_of X,[:L,L:] by WAYBEL_0:75;
A4: ex_sup_of D1,L & ex_sup_of D2,L by WAYBEL_0:75;
  thus (sup D1) "/\" (sup D2) = f.(sup D1,sup D2) by Def4
    .= f.(sup X) by A4,YELLOW_3:43
    .= sup (f.:X) by A2,A3
    .= sup (D1 "/\" D2) by Th32;
end;
