reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem Th12:
 for x being object holds
  x in (Partial_Intersection A1).n iff for k st k <= n holds x in A1.k
proof let x be object;
  defpred P[Nat] means (for k st k <= $1 holds x in A1.k) implies x in (
  Partial_Intersection A1).$1;
A1: for i st P[i] holds P[i+1]
  proof
    let i such that
A2: (for k st k <= i holds x in A1.k) implies x in (
    Partial_Intersection A1).i;
    assume for k st k <= i+1 holds x in A1.k;
    then
A3: ( for k st k <= i holds x in A1.k)& x in A1.(i+1) by NAT_1:12;
    (Partial_Intersection A1).(i+1) = (Partial_Intersection A1).i /\ A1.(
    i+1) by Def1;
    hence thesis by A2,A3,XBOOLE_0:def 4;
  end;
  thus x in (Partial_Intersection A1).n implies for k st k <= n holds x in A1.
  k
  proof
    assume
A4: x in (Partial_Intersection A1).n;
    for k st k <= n holds x in A1.k
    proof
A5:   Partial_Intersection A1 is non-ascending by Th10;
      let k such that
A6:   k <= n;
A7:   (Partial_Intersection A1).k c= A1.k by Th8;
      (Partial_Intersection A1).n c= (Partial_Intersection A1).k by A6,A5,
PROB_1:def 4;
      then (Partial_Intersection A1).n c= A1.k by A7;
      hence thesis by A4;
    end;
    hence thesis;
  end;
A8: P[0]
  proof
    assume for k st k <= 0 holds x in A1.k;
    then x in A1.0;
    hence thesis by Def1;
  end;
  for n holds P[n] from NAT_1:sch 2(A8,A1);
  hence thesis;
end;
