reserve M,N for non empty multMagma,
  f for Function of M, N;
reserve M for multMagma;
reserve N,K for multSubmagma of M;

theorem Th9:
  for F being Function st
  (for i being set st i in dom F holds F.i is stable Subset of M) holds
  meet F is stable Subset of M
proof
  let F be Function;
  assume A1: for i being set st i in dom F holds F.i is stable Subset of M;
  A2: for x being object st x in meet F holds x in the carrier of M
  proof
    let x be object;
    assume x in meet F; then
    A3: x in meet rng F by FUNCT_6:def 4;
    per cases;
    suppose dom F is empty; then
      F = {};
      hence thesis by A3,SETFAM_1:1;
    end;
    suppose dom F is not empty; then
      consider i be object such that
      A4: i in dom F by XBOOLE_0:def 1;
      meet rng F c= F.i by A4,FUNCT_1:3,SETFAM_1:3; then
      A5: x in F.i by A3;
      F.i is stable Subset of M by A1,A4;
      hence x in the carrier of M by A5;
    end;
  end;
  for v,w being Element of M st v in meet F & w in meet F holds v*w in meet F
  proof
    let v,w be Element of M;
    assume A6: v in meet F & w in meet F;
    per cases;
    suppose F = {}; then
      meet rng F = {} by SETFAM_1:1;
      hence thesis by A6,FUNCT_6:def 4;
    end;
    suppose A7: F <> {};
      A8: v in meet rng F & w in meet rng F by A6,FUNCT_6:def 4;
      for Y being set holds Y in rng F implies v*w in Y
      proof
        let Y be set;
        assume A9: Y in rng F; then
        A10: v in Y & w in Y by A8,SETFAM_1:def 1;
        consider i be object such that
        A11: i in dom F & Y = F.i by A9,FUNCT_1:def 3;
        Y is stable Subset of M by A1,A11;
        hence v*w in Y by A10,Def10;
      end; then
      v*w in meet rng F by A7,SETFAM_1:def 1;
      hence v*w in meet F by FUNCT_6:def 4;
    end;
  end;
  hence thesis by A2,Def10,TARSKI:def 3;
end;
