reserve X for set;
reserve UN for Universe;

theorem Th32:
  for I,a,b,u,v being Element of UN
  for x being UN-valued ManySortedSet of I st
  I = { a, b } & x.a = u & x.b = v holds
  disjoint-union x = [: u, { a } :] \/ [: v, { b }:]
  proof
    let I,a,b,u,v be Element of UN;
    let x be UN-valued ManySortedSet of I;
    assume that
A1: I = { a, b } and
A2: x.a = u and
A3: x.b = v;
A4: dom disjoin x = dom x & for o be object st o in dom x holds
      (disjoin x).o = [: x.o , {o} :] by CARD_3:def 3;
A5: dom x = { a, b } by A1,PARTFUN1:def 2;
    then a in dom x & b in dom x by TARSKI:def 2;
    then
A6: (disjoin x).a = [: x.a, { a } :] & (disjoin x).b = [: x.b, { b }:]
      by CARD_3:def 3;
    set S = { [: x.a, { a } :], [: x.b, { b }:] };
    rng disjoin x = { [: x.a, { a } :], [: x.b, { b }:] }
    proof
      now
        now
          let o be object;
          assume o in rng disjoin x;
          then consider o9 be object such that
A7:       o9 in dom disjoin x and
A8:       o = (disjoin x).o9 by FUNCT_1:def 3;
          o9 = a or o9 = b by A7,A4,A5,TARSKI:def 2;
          hence o in S by A8,A6,TARSKI:def 2;
        end;
        hence rng disjoin x c= S;
        now
          let o be object;
          assume o in S; then
A9:       o = (disjoin x).a or o = (disjoin x).b by A6,TARSKI:def 2;
          a in dom disjoin x & b in dom disjoin x by A4,A5,TARSKI:def 2;
          hence o in rng disjoin x by A9,FUNCT_1:def 3;
        end;
        hence S c= rng disjoin x;
      end;
      hence thesis;
    end;
    hence thesis by A2,A3,ZFMISC_1:75;
  end;
