reserve X for set,
        A for Subset of X,
        R,S for Relation of X;

theorem
  [:X \ A,X:] \/ [:X,A:] = [:A,A:] \/ [:X \ A,X:]
  proof
A1: [:X \ A,X:] \/ [:X,A:] c= [:A,A:] \/ [:X\A,X:]
    proof
      let x be object;
      assume
A2:   x in [:X \ A,X:] \/ [:X,A:];
      [:X \ A,X:] \/ [:X,A:] c= [:X,X:] by Th2;
      then consider a,b be object such that
A3:   a in X and
A4:   b in X and
A5:   x = [a,b] by A2,ZFMISC_1:def 2;
      per cases;
      suppose
A6:     a in A;
        x in [:X,A:]
        proof
          assume not x in [:X,A:];
          then x in [:X \ A,X:] by A2,XBOOLE_0:def 3;
          then a in X \ A & b in X by A5,ZFMISC_1:87;
          hence contradiction by A6,XBOOLE_0:def 5;
        end;
        then b in A by A5,ZFMISC_1:87; then
A7:     x in [:A,A:] by A6,A5,ZFMISC_1:87;
        [:A,A:] c= [:A,A:] \/ [:X\A,X:] by XBOOLE_1:7;
        hence thesis by A7;
      end;
      suppose not a in A;
        then a in X \ A by A3,XBOOLE_0:def 5; then
A8:     x in [:X\A,X:] by A4,A5,ZFMISC_1:87;
        [:X\A,X:] c= [:X\A,X:] \/ [:A,A:] by XBOOLE_1:7;
        hence thesis by A8;
      end;
    end;
    [:A,A:] c= [:X,A:] by ZFMISC_1:95;
    hence thesis by A1,XBOOLE_1:13;
  end;
