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;
