
theorem
for X,Y being set st Y c= X holds (X \ Y) \/ Y = X
proof
let X,Y be set;
assume AS: Y c= X;
A: now let o be object;
   assume B: o in X;
   per cases;
   suppose o in Y;
     hence o in (X \ Y) \/ Y by XBOOLE_0:def 3;
     end;
   suppose not o in Y;
     then o in X \ Y by B,XBOOLE_0:def 5;
     hence o in (X \ Y) \/ Y by XBOOLE_0:def 3;
     end;
   end;
now let o be object;
  assume B: o in (X \ Y) \/ Y;
  now assume C: not o in X;
    then not o in (X \ Y);
    then o in Y by B,XBOOLE_0:def 3;
    hence contradiction by AS,C;
    end;
  hence o in X;
  end;
hence thesis by A,TARSKI:2;
end;
