reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;
reserve X, Y, Z for ManySortedSet of I;

theorem
  for M be non-empty finite-yielding ManySortedSet of I st for A, B be
  ManySortedSet of I st A in M & B in M holds A c= B or B c= A ex m be
ManySortedSet of I st m in M & for K be ManySortedSet of I st K in M holds m c=
  K
proof
  let M be non-empty finite-yielding ManySortedSet of I such that
A1: for A, B be ManySortedSet of I st A in M & B in M holds A c= B or B c= A;
  defpred Q[object,object] means
   ex D being set st D = $2 &
   $2 in M.$1 & for D9 be set st D9 in M.$1 holds D c= D9;
A2: for i being object st i in I ex j be object st Q[i,j]
  proof
    let i be object;
    assume
A3: i in I;
    M.i is c=-linear
    proof
      consider c9 be ManySortedSet of I such that
A4:   c9 in M by PBOOLE:134;
      consider b9 be ManySortedSet of I such that
A5:   b9 in M by PBOOLE:134;
      let B9, C9 be set such that
A6:   B9 in M.i and
A7:   C9 in M.i;
A8:   dom (i .--> B9) = {i};
      set qc = c9 +* (i.-->C9);
      dom qc = I by A3,PZFMISC1:1;
      then reconsider qc as ManySortedSet of I by PARTFUN1:def 2,RELAT_1:def 18
;
A9:   dom (i .--> C9) = {i};
      i in {i} by TARSKI:def 1;
      then
A10:  qc.i = (i .--> C9).i by A9,FUNCT_4:13
        .= C9 by FUNCOP_1:72;
A11:  qc in M
      proof
        let j be object such that
A12:    j in I;
        now
          per cases;
          case
            j = i;
            hence thesis by A7,A10;
          end;
          case
            j <> i;
            then not j in dom (i .--> C9) by TARSKI:def 1;
            then qc.j = c9.j by FUNCT_4:11;
            hence thesis by A4,A12;
          end;
        end;
        hence thesis;
      end;
      set qb = b9 +* (i.-->B9);
      dom qb = I by A3,PZFMISC1:1;
      then reconsider qb as ManySortedSet of I by PARTFUN1:def 2,RELAT_1:def 18
;
      assume
A13:  not B9 c= C9;
      i in {i} by TARSKI:def 1;
      then
A14:  qb.i = (i .--> B9).i by A8,FUNCT_4:13
        .= B9 by FUNCOP_1:72;
      qb in M
      proof
        let j be object such that
A15:    j in I;
        now
          per cases;
          case
            j = i;
            hence thesis by A6,A14;
          end;
          case
            j <> i;
            then not j in dom (i .--> B9) by TARSKI:def 1;
            then qb.j = b9.j by FUNCT_4:11;
            hence thesis by A5,A15;
          end;
        end;
        hence thesis;
      end;
      then qb c= qc or qc c= qb by A1,A11;
      hence thesis by A3,A13,A14,A10;
    end;
    then consider m9 be set such that
A16: m9 in M.i & for D9 be set st D9 in M.i holds m9 c= D9 by A3,FINSET_1:11;
    take m9;
    thus thesis by A16;
  end;
  consider m be ManySortedSet of I such that
A17: for i being object st i in I holds Q[i,m.i] from PBOOLE:sch 3(A2);
  take m;
  thus m in M
  proof
    let i be object;
    assume i in I;
     then Q[i,m.i] by A17;
    hence thesis;
  end;
  thus for C be ManySortedSet of I st C in M holds m c= C
  proof
    let C be ManySortedSet of I such that
A18: C in M;
    let i be object;
    assume
A19: i in I;
    then
A20:  C.i in M.i by A18;
    Q[i,m.i] by A17,A19;
    hence thesis by A20;
  end;
end;
