reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve P,P1,P2 for Probability of Sigma;

theorem Th32:
  for P,A,A1,A2,A3 st A1 misses A2 & A3 = (A1 \/ A2)` & 0 < P.A1 &
0 < P.A2 & 0 < P.A3 holds P.A = (P.|.A1.A * P.A1) + (P.|.A2.A * P.A2) + (P.|.A3
  .A * P.A3)
proof
  let P,A,A1,A2,A3;
  assume that
A1: A1 misses A2 and
A2: A3 = (A1 \/ A2)` and
A3: 0 < P.A1 and
A4: 0 < P.A2 and
A5: 0 < P.A3;
A6: A /\ A1 misses A /\ A2 by A1,XBOOLE_1:76;
  (A1 \/ A2) misses A3 by A2,SUBSET_1:24;
  then
A7: A /\ (A1 \/ A2) misses A /\ A3 by XBOOLE_1:76;
A8: A1 \/ A2 \/ A3 = [#]Omega by A2,SUBSET_1:10
    .= Omega;
  (P.|.A1.A * P.A1) + (P.|.A2.A * P.A2) + (P.|.A3.A * P.A3) = P.(A /\ A1)
  + (P.|.A2.A * P.A2) + (P.|.A3.A * P.A3) by A3,Th29
    .= P.(A /\ A1) + P.(A /\ A2) + (P.|.A3.A * P.A3) by A4,Th29
    .= P.(A /\ A1) + P.(A /\ A2) + P.(A /\ A3) by A5,Th29
    .= P.((A /\ A1) \/ (A /\ A2)) + P.(A /\ A3) by A6,PROB_1:def 8
    .= P.(A /\ (A1 \/ A2)) + P.(A /\ A3) by XBOOLE_1:23
    .= P.((A /\ (A1 \/ A2)) \/ (A /\ A3)) by A7,PROB_1:def 8
    .= P.(A /\ Omega) by A8,XBOOLE_1:23
    .= P.A by XBOOLE_1:28;
  hence thesis;
end;
