 reserve R for 1-sorted;
 reserve X,Y for Subset of R;

theorem LemmaSet:
  X c= Y iff X` \/ Y = [#]R
  proof
    thus X c= Y implies X` \/ Y = [#]R
    proof
      assume X c= Y; then
      X \ Y = {} by XBOOLE_1:37; then
      (X \ Y)` = [#]R;
      hence thesis by SUBSET_1:14;
    end;
    assume X` \/ Y = [#]R; then
    (X \ Y)`` = ({}R)`` by SUBSET_1:14;
    hence thesis by XBOOLE_1:37;
  end;
