
theorem Th14:
  for X being set, A being SetSequence of X, B being Subset of X
  st (for n being Nat holds A.n = B) holds lim_sup A = B
proof
  let X be set, A be SetSequence of X, B be Subset of X;
  assume
A1: for n being Nat holds A.n = B;
  thus lim_sup A c= B
  proof
    let x be object;
    assume x in lim_sup A;
    then ex k being Nat st x in A.(0 qua Nat+k) by Th5;
    hence thesis by A1;
  end;
  thus B c= lim_sup A
  proof
    let x be object;
    assume
A2: x in B;
    for m being Nat ex k being Nat st x in A.(m+k)
    proof
      let m be Nat;
      take 0;
      thus thesis by A1,A2;
    end;
    hence thesis by Th5;
  end;
end;
