
theorem Th17:
  for H being complete distributive LATTICE, a being Element of H
  for X being finite Subset of H holds a "/\" preserves_sup_of X
proof
  let H be complete distributive LATTICE, a be Element of H, X be finite
  Subset of H;
  assume ex_sup_of X,H;
  thus ex_sup_of a "/\".:X,H by YELLOW_0:17;
  thus sup (a"/\".:X) = "\/"({a"/\"y where y is Element of H: y in X},H) by
WAYBEL_1:61
    .= sup({a} "/\" X) by YELLOW_4:42
    .= a "/\" sup X by Th15
    .= a"/\".sup X by WAYBEL_1:def 18;
end;
