reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;

theorem
  for X being Element of bool M for i, x being set st
  i in I & x in ((id (bool M))..X).i
  ex Y being finite-yielding Element of bool M st
  Y c= X & x in ((id (bool M))..Y).i
proof
  let X be Element of bool M, i, x be set such that
A1: i in I and
A2: x in ((id (bool M))..X).i;
A3: x in X.i by A2,Th8;
  set Y = (I --> {}) +* (i.-->{x});
  dom Y = I by A1,PZFMISC1:1;
  then reconsider Y as ManySortedSet of I by PARTFUN1:def 2,RELAT_1:def 18;
A4: dom (i .--> {x}) = {i};
  i in {i} by TARSKI:def 1;
  then
A5: Y.i = (i .--> {x}).i by A4,FUNCT_4:13
    .= {x} by FUNCOP_1:72;
  X in bool M by MSSUBFAM:12;
  then X c= M by MBOOLEAN:18;
  then
A6: X.i c= M.i by A1;
  Y is Element of bool M
  proof
    let j be object such that
A7: j in I;
    now
      per cases;
      case
A8:     j = i;
        then {x} c= M.j by A3,A6,ZFMISC_1:31;
        hence thesis by A5,A7,A8,MBOOLEAN:def 1;
      end;
      case
        j <> i;
        then not j in dom (i .--> {x}) by TARSKI:def 1;
        then Y.j = (I --> {}).j by FUNCT_4:11;
        then Y.j = {};
        then Y.j c= M.j;
        hence thesis by A7,MBOOLEAN:def 1;
      end;
    end;
    hence thesis;
  end;
  then reconsider Y as Element of bool M;
  Y is finite-yielding
  proof
    let j be object such that
 j in I;
    now
      per cases;
      case
        j = i;
        hence thesis by A5;
      end;
      case
        j <> i;
        then not j in dom (i .--> {x}) by TARSKI:def 1;
        then Y.j = (I --> {}).j by FUNCT_4:11;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  then reconsider Y as finite-yielding Element of bool M;
  take Y;
  thus Y c= X
  proof
    let j be object such that
 j in I;
    now
      per cases;
      case
        j = i;
        hence thesis by A3,A5,ZFMISC_1:31;
      end;
      case
        j <> i;
        then not j in dom (i .--> {x}) by TARSKI:def 1;
        then Y.j = (I --> {}).j by FUNCT_4:11;
        then Y.j = {};
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  x in Y.i by A5,TARSKI:def 1;
  hence thesis by Th8;
end;
