
theorem
  for T being set, F being SetSequence of T st F is non-ascending holds
  lim_inf F = meet F
proof
  let T be set, F be SetSequence of T;
  assume
A1: F is non-ascending;
  thus lim_inf F c= meet F
  proof
    let x be object;
    assume x in lim_inf F;
    then consider n being Nat such that
A2: for k being Nat holds x in F.(n+k) by Th4;
    for k being Nat st k > n holds x in F.k
    proof
      let k be Nat;
      assume k > n;
      then consider h being Nat such that
A3:   k = n + h by NAT_1:10;
      thus thesis by A2,A3;
    end;
    hence thesis by A1,Th19;
  end;
  thus thesis by Th7;
end;
