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 Th16:
  x in (Partial_Diff_Union A1).n iff x in A1.n & for k st k < n
  holds not x in A1.k
proof
  thus x in (Partial_Diff_Union A1).n implies x in A1.n & for k st k < n holds
  not x in A1.k
  proof
    assume
A1: x in (Partial_Diff_Union A1).n;
    now
      per cases by NAT_1:6;
      case
        n = 0;
        hence thesis by A1,Def3;
      end;
      case
        ex n1 being Nat st n = n1 + 1;
        then consider n1 being Nat such that
A2:     n = n1+1;
A3:     x in A1.(n1+1) \ (Partial_Union A1).n1 by A1,A2,Def3;
        then
A4:     not x in (Partial_Union A1).n1 by XBOOLE_0:def 5;
        for k st k < n holds not x in A1.k
        proof
          let k;
          assume k < n;
          then k <= n1 by A2,NAT_1:13;
          hence thesis by A4,Th13;
        end;
        hence thesis by A2,A3,XBOOLE_0:def 5;
      end;
    end;
    hence thesis;
  end;
  assume that
A5: x in A1.n and
A6: for k st k < n holds not x in A1.k;
  now
    per cases by NAT_1:6;
    case
      n = 0;
      hence thesis by A5,Def3;
    end;
    case
      ex n1 being Nat st n = n1 + 1;
      then consider n1 being Nat such that
A7:   n = n1+1;
      for k st k <= n1 holds not x in A1.k
      proof
        let k;
        assume k <= n1;
        then k < n1+1 by NAT_1:13;
        hence thesis by A6,A7;
      end;
      then not x in (Partial_Union A1).n1 by Th13;
      then x in A1.(n1+1) \ (Partial_Union A1).n1 by A5,A7,XBOOLE_0:def 5;
      hence thesis by A7,Def3;
    end;
  end;
  hence thesis;
end;
