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 st A c= B holds
P . A <= P . B
let Sigma be SigmaField of Omega; :: thesis: for A, B being Event of Sigma
for P being Probability of Sigma st A c= B holds
P . A <= P . B
let A, B be Event of Sigma; :: thesis: for P being Probability of Sigma st A c= B holds
P . A <= P . B
let P be Probability of Sigma; :: thesis: ( A c= B implies P . A <= P . B )
assume A c= B ; :: thesis: P . A <= P . B
then P . (B \ A) = (P . B) - (P . A) by Th33;
then 0 <= (P . B) - (P . A) by Def8;
then 0 + (P . A) <= P . B by XREAL_1:19;
hence P . A <= P . B ; :: thesis: verum