let Omega be non empty set ; :: thesis: for Sigma being SigmaField of Omega
for A, B being Event of Sigma
for P being Probability of Sigma holds ((P . A) + (P . B)) - 1 <= P . (A /\ B)
let Sigma be SigmaField of Omega; :: thesis: for A, B being Event of Sigma
for P being Probability of Sigma holds ((P . A) + (P . B)) - 1 <= P . (A /\ B)
let A, B be Event of Sigma; :: thesis: for P being Probability of Sigma holds ((P . A) + (P . B)) - 1 <= P . (A /\ B)
let P be Probability of Sigma; :: thesis: ((P . A) + (P . B)) - 1 <= P . (A /\ B)
((P . A) + (P . B)) - (P . (A /\ B)) = P . (A \/ B) by PROB_1:38;
then ((P . A) + (P . B)) - (P . (A /\ B)) <= 1 by PROB_1:35;
then (P . A) + (P . B) <= (P . (A /\ B)) + 1 by XREAL_1:20;
hence ((P . A) + (P . B)) - 1 <= P . (A /\ B) by XREAL_1:20; :: thesis: verum