
theorem
  for W being with_non-empty_element set holds
  W-CL-opp_category, W-CL_category are_anti-isomorphic_under W UpperAdj
proof
  let W be with_non-empty_element set;
  set A1 = W-INF_category, A2 = W-SUP_category;
  reconsider B1 = W-CL_category as non empty subcategory of A1 by ALTCAT_4:36;
  reconsider B2 = W-CL-opp_category as non empty subcategory of A2
  by ALTCAT_4:36;
  B2, B1 are_anti-isomorphic_under (W LowerAdj)" by Th54,YELLOW20:51;
  hence thesis by Th18;
end;
