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;
