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 Th25:
  for A,B,P st A,B are_independent_respect_to P holds A,([#] Sigma
  \ B) are_independent_respect_to P
proof
  let A,B,P;
  assume A,B are_independent_respect_to P;
  then
A1: P.(A /\ B) = P.A * P.B;
  P.(A /\ ([#] Sigma \ B)) = P.(A /\ B`) .= P.(A \ B) by SUBSET_1:13
    .= P.(A \ (A /\ B)) by XBOOLE_1:47
    .= P.A * 1 - P.A * P.B by A1,PROB_1:33,XBOOLE_1:17
    .= P.A * (1 - P.B)
    .= P.A * P.([#] Sigma \ B) by PROB_1:32;
  hence thesis;
end;
