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 Th19:
  Partial_Union (Partial_Diff_Union A1) = Partial_Union A1
proof
  for n being Nat holds (Partial_Union (Partial_Diff_Union A1))
  .n = (Partial_Union A1).n
  proof
    set A2 = Partial_Diff_Union A1;
    defpred P[set] means (Partial_Union A2).$1 = (Partial_Union A1).$1;
A1: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
A2:   (Partial_Union A2).k = (Partial_Union A1).k;
      thus (Partial_Union A2).(k+1) = A2.(k+1) \/ (Partial_Union A2).k by Def2
        .= (A1.(k+1) \ (Partial_Union A1).k) \/ (Partial_Union A1).k by A2,Def3
        .= A1.(k+1) \/ (Partial_Union A1).k by XBOOLE_1:39
        .= (Partial_Union A1).(k+1) by Def2;
    end;
    (Partial_Union (Partial_Diff_Union A1)).0 = A2.0 by Def2
      .= A1.0 by Def3
      .= (Partial_Union A1).0 by Def2;
    then
A3: P[0];
    thus for k being Nat holds P[k] from NAT_1:sch 2(A3,A1 );
  end;
  hence thesis;
end;
