let Omega1, Omega2 be non empty set ; :: thesis: for S1 being SigmaField of Omega1
for S2 being SigmaField of Omega2
for P being Probability of S1
for F being random_variable of S1,S2
for y being set st y in S2 holds
(probability (F,P)) . y = P . (F " y)

let S1 be SigmaField of Omega1; :: thesis: for S2 being SigmaField of Omega2
for P being Probability of S1
for F being random_variable of S1,S2
for y being set st y in S2 holds
(probability (F,P)) . y = P . (F " y)

let S2 be SigmaField of Omega2; :: thesis: for P being Probability of S1
for F being random_variable of S1,S2
for y being set st y in S2 holds
(probability (F,P)) . y = P . (F " y)

let P be Probability of S1; :: thesis: for F being random_variable of S1,S2
for y being set st y in S2 holds
(probability (F,P)) . y = P . (F " y)

let F be random_variable of S1,S2; :: thesis: for y being set st y in S2 holds
(probability (F,P)) . y = P . (F " y)

let y be set ; :: thesis: ( y in S2 implies (probability (F,P)) . y = P . (F " y) )
assume A1: y in S2 ; :: thesis: (probability (F,P)) . y = P . (F " y)
thus (probability (F,P)) . y = (image_measure (F,(P2M P))) . y by Th13
.= P . (F " y) by ; :: thesis: verum