reserve x, y for object, I for set,
  A, B, X, Y for ManySortedSet of I;

theorem :: ZFMISC_1:94
  for Z be ManySortedSet of I for A be non-empty ManySortedSet of I
holds (for X be ManySortedSet of I st X in A holds X c= Z) implies union A c= Z
proof
  let Z be ManySortedSet of I, A be non-empty ManySortedSet of I;
  assume
A1: for X be ManySortedSet of I st X in A holds X c= Z;
  let i be object such that
A2: i in I;
  for X9 be set st X9 in A.i holds X9 c= Z.i
  proof
    consider M be ManySortedSet of I such that
A3: M in A by PBOOLE:134;
    let X9 be set such that
A4: X9 in A.i;
    dom (M +* (i .--> X9)) = I by A2,Lm1;
    then reconsider K = M +* (i .--> X9) as ManySortedSet of I by
PARTFUN1:def 2,RELAT_1:def 18;
A5: dom (i .--> X9) = {i};
    i in {i} by TARSKI:def 1;
    then
A6: K.i = (i .--> X9).i by A5,FUNCT_4:13
      .= X9 by FUNCOP_1:72;
    K in A
    proof
      let j be object such that
A7:   j in I;
      now
        per cases;
        case
          j = i;
          hence thesis by A4,A6;
        end;
        case
          j <> i;
          then not j in dom (i .--> X9) by TARSKI:def 1;
          then K.j = M.j by FUNCT_4:11;
          hence thesis by A3,A7;
        end;
      end;
      hence thesis;
    end;
    then K c= Z by A1;
    hence thesis by A2,A6;
  end;
  then union (A.i) c= Z.i by ZFMISC_1:76;
  hence thesis by A2,Def2;
end;
