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
  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;
