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 A,B,P st 0 < P.B holds 1 - P.([#] Sigma \ A)/P.B <= P.|.B.A
proof
  let A,B,P;
  assume
A1: 0 < P.B;
  P.B + P.A - 1 <= P.(A /\ B) by Th15;
  then P.B + -(1 - P.A) <= P.(A /\ B);
  then P.B + -P.([#] Sigma \ A) <= P.(A /\ B) by PROB_1:32;
  then (P.B + -P.([#] Sigma \ A))/P.B <= P.(A /\ B)/P.B by A1,XREAL_1:72;
  then (P.B - P.([#] Sigma \ A))/P.B <= P.|.B.A by A1,Def6;
  then P.B/P.B - P.([#] Sigma \ A)/P.B <= P.|.B.A by XCMPLX_1:120;
  hence thesis by A1,XCMPLX_1:60;
end;
