
theorem Th40:
  for L being up-complete Semilattice st for I1, I2 being Ideal of
L holds (sup I1) "/\" (sup I2) = sup (I1 "/\" I2) holds for D1, D2 be directed
  non empty Subset of L holds (sup D1) "/\" (sup D2) = sup (D1 "/\" D2)
proof
  let L be up-complete Semilattice such that
A1: for I1, I2 being Ideal of L holds (sup I1) "/\" (sup I2) = sup (I1
  "/\" I2);
  let D1, D2 be directed non empty Subset of L;
A2: ex_sup_of D2,L by WAYBEL_0:75;
A3: ex_sup_of (downarrow D1 "/\" downarrow D2),L by WAYBEL_0:75;
A4: ex_sup_of D1 "/\" D2,L by WAYBEL_0:75;
  ex_sup_of D1,L by WAYBEL_0:75;
  hence (sup D1) "/\" (sup D2) = (sup downarrow D1) "/\" (sup D2) by
WAYBEL_0:33
    .= (sup downarrow D1) "/\" (sup downarrow D2) by A2,WAYBEL_0:33
    .= sup (downarrow D1 "/\" downarrow D2) by A1
    .= sup downarrow ((downarrow D1) "/\" (downarrow D2)) by A3,WAYBEL_0:33
    .= sup downarrow (D1 "/\" D2) by YELLOW_4:62
    .= sup (D1 "/\" D2) by A4,WAYBEL_0:33;
end;
