reserve X,Y,Z,x,y,z for set;

theorem
  rng Total X = X
proof
  for x being object holds x in X iff ex y being object st [y,x] in Total X
  proof
    let x be object;
    thus x in X implies ex y being object st [y,x] in Total X
    proof
      assume
A1:   x in X;
      take x;
      [x,x] in [:X,X:] by A1,ZFMISC_1:87;
      hence thesis by EQREL_1:def 1;
    end;
    thus thesis by ZFMISC_1:87;
  end;
  hence thesis by XTUPLE_0:def 13;
end;
