theorem Th14:
  Intersection Partial_Intersection A1 = Intersection A1
proof
  thus Intersection Partial_Intersection A1 c= Intersection A1
  proof
    let x be object;
    assume
A1: x in Intersection Partial_Intersection A1;
    for n be Nat holds x in A1.n
    proof
      let n be Nat;
      x in (Partial_Intersection A1).n by A1,PROB_1:13;
      hence thesis by Th12;
    end;
    hence thesis by PROB_1:13;
  end;
  let x be object;
  assume
A2: x in Intersection A1;
  for n be Nat holds x in (Partial_Intersection A1).n
  proof
    let n be Nat;
    for k st k <= n holds x in A1.k
    by A2,PROB_1:13;
    hence thesis by Th12;
  end;
  hence thesis by PROB_1:13;
end;
