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 A,B are_independent_respect_to P & P.A < 1 & P.B < 1
  holds P.(A \/ B) < 1
proof
A1: now
    let r,r1;
    assume 0 < r1;
    then - r1 < -0 by XREAL_1:24;
    then r + - r1 < r + 0 by XREAL_1:6;
    hence r - r1 < r;
  end;
  let P,A,B;
  assume that
A2: A,B are_independent_respect_to P and
A3: P.A < 1 & P.B < 1;
A4: ([#] Sigma \ A),([#] Sigma \ B) are_independent_respect_to P by A2,Th26;
A5: 0 < P.([#] Sigma \ A) & 0 < P.([#] Sigma \ B) by A3,Th17;
  P.(A \/ B) = 1 - P.([#] Sigma \ (A \/ B)) by Th16
    .= 1 - P.(([#] Sigma \ A) /\ ([#] Sigma \ B)) by XBOOLE_1:53
    .= 1 - P.([#] Sigma \ A) * P.([#] Sigma \ B) by A4;
  hence thesis by A5,A1,XREAL_1:129;
end;
