theorem
  [:X,Y:]~ = [:Y,X:]
proof
  let x,y be object;
  thus [x,y] in [:X,Y:]~ implies [x,y] in [:Y,X:]
  proof
    assume [x,y] in [:X,Y:]~;
    then [y,x] in [:X,Y:] by RELAT_1:def 7;
    then y in X & x in Y by ZFMISC_1:87;
    hence thesis by ZFMISC_1:87;
  end;
  assume [x,y] in [:Y,X:];
  then y in X & x in Y by ZFMISC_1:87;
  then [y,x] in [:X,Y:] by ZFMISC_1:87;
  hence thesis by RELAT_1:def 7;
end;
