
theorem
  for X being set, F being SetSequence of X holds lim_inf F c= lim_sup F
proof
  let X be set, F be SetSequence of X;
  let x be object;
  assume x in lim_inf F;
  then consider n be Nat such that
A1: for k being Nat holds x in F.(n+k) by Th4;
  now
    let k be Nat;
    x in F.(n+k) by A1;
    hence ex n being Nat st x in F.(k+n);
  end;
  hence thesis by Th5;
end;
