
theorem Th19:
  for T being set, F being SetSequence of T, x being object st F is
non-ascending & ex k being Nat st for n being Nat st n >
  k holds x in F.n holds x in meet F
proof
  let T be set, F be SetSequence of T, x be object;
  assume
A1: F is non-ascending;
  given k being Nat such that
A2: for n being Nat st n > k holds x in F.n;
  k + 1 > k by NAT_1:13;
  then
A3: x in F.(k + 1) by A2;
  assume not x in meet F;
  then not x in meet rng F by FUNCT_6:def 4;
  then consider Y being set such that
A4: Y in rng F and
A5: not x in Y by SETFAM_1:def 1;
  consider y being object such that
A6: y in dom F and
A7: Y = F.y by A4,FUNCT_1:def 3;
  reconsider y as Nat by A6;
  per cases;
  suppose
    y > k;
    hence thesis by A2,A5,A7;
  end;
  suppose
    y <= k;
    then F.k c= F.y by A1,PROB_1:def 4;
    then
A8: not x in F.k by A5,A7;
    F.(k + 1) c= F.k by A1;
    hence thesis by A3,A8;
  end;
end;
