
theorem LM1:
  for M be non empty set,
      x be sequence of M st rng x is finite
  ex z be Element of M st x"{z} c= NAT & x"{z} is infinite
  proof
    let M be non empty set,
        x be sequence of M;
    assume
    A1: rng x is finite;
    assume
    A2: not (ex z be Element of M st x"{z} c= NAT & x"{z} is infinite);
    deffunc X(object) = x"{$1};
    set K = {X(w) where w is Element of M: w in rng x};
    A3: K is finite from FRAENKEL:sch 21(A1);
    for Y be set st Y in K holds Y is finite
    proof
      let Y be set;
      assume Y in K; then
      consider w be Element of M such that
      A4: Y = x"{w} & w in rng x;
      thus Y is finite by A2,A4;
    end; then
    A5: union K is finite by A3,FINSET_1:7;
    dom x c= union K
    proof
      let s be object;
      assume
  A6: s in dom x; then
      reconsider sn = s as Element of NAT;
      reconsider w = x.sn as Element of M;
      w in rng x by A6,FUNCT_1:3; then
  A7: x"{w} in K;
      w in {w} by TARSKI:def 1; then
      s in x"{w} by A6,FUNCT_1:def 7;
      hence s in union K by A7,TARSKI:def 4;
    end;
    hence contradiction by A5,FUNCT_2:def 1;
  end;
