reserve X for set;
reserve UN for Universe;

theorem
  for I being Element of UN,
      x being UN-valued ManySortedSet of I
  holds disjoint-union x is Subset of [: union rng x,I :]
  proof
    let I be Element of UN;
    let x be UN-valued ManySortedSet of I;
    union rng disjoin x c= [: union rng x,I :]
    proof
      let o be object;
      assume o in union rng disjoin x;
      then consider o9 be set such that
A1:   o in o9 and
A2:   o9 in rng disjoin x by TARSKI:def 4;
      consider y be object such that
A3:   y in dom disjoin x and
A4:   o9 = (disjoin x).y by A2,FUNCT_1:def 3;
A5:   y in dom x by A3,CARD_3:def 3;
A6:   y in I by A3,PARTFUN1:def 2;
A7:   o in [: x.y,{y} :] by A1,A4,A5,CARD_3:def 3;
      now
        let o be object;
        assume o in [: x.y,{y} :];
        then consider o1,o2 be object such that
A8:     o1 in x.y and
A9:     o2 in {y} and
A10:     o = [o1,o2] by ZFMISC_1:def 2;
        now
          x.y in rng x by A5,FUNCT_1:def 3;
          hence o1 in union rng x by A8,TARSKI:def 4;
          thus o2 in I by A6,A9,TARSKI:def 1;
        end;
        hence o in [: union rng x,I:] by A10,ZFMISC_1:def 2;
      end;
      hence thesis by A7;
    end;
    hence thesis;
  end;
