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 Th13:
  x in (Partial_Union A1).n iff ex k st k <= n & x in A1.k
proof
  defpred P[Nat] means (x in (Partial_Union A1).$1 implies ex k st k <= $1 & x
  in A1.k);
A1: for i st P[i] holds P[i+1]
  proof
    let i such that
A2: x in (Partial_Union A1).i implies ex k st k <= i & x in A1.k;
    assume
A3: x in (Partial_Union A1).(i+1);
A4: (Partial_Union A1).(i+1) = (Partial_Union A1).i \/ A1.(i+1) by Def2;
    now
      per cases by A3,A4,XBOOLE_0:def 3;
      case
        x in (Partial_Union A1).i;
        then consider k such that
A5:     k <= i & x in A1.k by A2;
        take k;
        thus thesis by A5,NAT_1:12;
      end;
      case
        x in A1.(i+1);
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A6: P[0]
  proof
    assume
A7: x in (Partial_Union A1).0;
    take 0;
    thus thesis by A7,Def2;
  end;
  for n holds P[n] from NAT_1:sch 2(A6,A1);
  hence x in (Partial_Union A1).n implies ex k st k <= n & x in A1.k;
  given i such that
A8: i <= n and
A9: x in A1.i;
  A1.i c= (Partial_Union A1).i by Th9;
  then
A10: x in (Partial_Union A1).i by A9;
A11: Partial_Union A1 is non-descending by Th11;
  (Partial_Union A1).i c= (Partial_Union A1).n by A8,A11,PROB_1:def 5;
  hence thesis by A10;
end;
