theorem
  P.([#] Sigma \ A) < 1 iff 0 < P.A
proof
  thus P.([#] Sigma \ A) < 1 implies 0 < P.A
  proof
    assume P.([#] Sigma \ A) < 1;
    then 1 - P.A < 1 by PROB_1:32;
    then 1 + - P.A < 1;
    then - P.A < 1 - 1 by XREAL_1:20;
    hence thesis;
  end;
  assume 0 < P.A;
  then 0 < 1 - P.([#] Sigma \ A) by Th16;
  then P.([#] Sigma \ A) + 0 < 1 by XREAL_1:20;
  hence thesis;
end;
