
theorem Th26:
  for L be complete Boolean LATTICE for X,Y be Subset of L st X c=
  ATOM L & Y c= ATOM L holds X c= Y iff sup X <= sup Y
proof
  let L be complete Boolean LATTICE;
  let X,Y be Subset of L;
  assume that
A1: X c= ATOM L and
A2: Y c= ATOM L;
  thus X c= Y implies sup X <= sup Y
  proof
A3: ex_sup_of X,L & ex_sup_of Y,L by YELLOW_0:17;
    assume X c= Y;
    hence thesis by A3,YELLOW_0:34;
  end;
  thus sup X <= sup Y implies X c= Y
  proof
    assume
A4: sup X <= sup Y;
    thus X c= Y
    proof
      let z be object;
      assume
A5:   z in X;
      then reconsider z1 = z as Element of L;
      sup X is_>=_than X by YELLOW_0:32;
      then z1 <= sup X by A5,LATTICE3:def 9;
      then
A6:   z1 <= sup Y by A4,ORDERS_2:3;
      z1 is atom by A1,A5,Def2;
      hence thesis by A2,A6,Th25;
    end;
  end;
end;
