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
  for P,A,B st 0 < P.B holds (P.|.B.A = P.A iff A,B
  are_independent_respect_to P)
proof
  let P,A,B;
  assume
A1: 0 < P.B;
  thus P.|.B.A = P.A implies A,B are_independent_respect_to P
  proof
    assume P.|.B.A = P.A;
    then P.(A /\ B)/P.B * P.B = P.A * P.B by A1,Def6;
    then P.(A /\ B) = P.A * P.B by A1,XCMPLX_1:87;
    hence thesis;
  end;
  assume A,B are_independent_respect_to P;
  then P.(A /\ B) * (P.B)" = P.A * P.B * (P.B)";
  then P.(A /\ B) * (P.B)" = P.A * (P.B * (P.B)");
  then P.(A /\ B) * (P.B)" = P.A * 1 by A1,XCMPLX_0:def 7;
  then P.(A /\ B)/P.B = P.A by XCMPLX_0:def 9;
  hence thesis by A1,Def6;
end;
