
theorem Th15:
  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_inf 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_inf A c= B
  proof
    let x be object;
    assume x in lim_inf A;
    then consider m being Nat such that
A2: for k being Nat holds x in A.(m+k) by Th4;
    x in A.(m+(0 qua Nat)) by A2;
    hence thesis by A1;
  end;
  thus B c= lim_inf A
  proof
    let x be object;
    assume
A3: x in B;
    ex m being Nat st for k being Nat holds x in A.( m+k)
    proof
      take 0;
      let k be Nat;
      thus thesis by A1,A3;
    end;
    hence thesis by Th4;
  end;
end;
