let Omega be non empty set ; :: thesis: for Sigma being SigmaField of Omega
for P being Probability of Sigma
for A, B, C being Event of Sigma holds
( A,B,C are_independent_respect_to P iff ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) )
let Sigma be SigmaField of Omega; :: thesis: for P being Probability of Sigma
for A, B, C being Event of Sigma holds
( A,B,C are_independent_respect_to P iff ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) )
let P be Probability of Sigma; :: thesis: for A, B, C being Event of Sigma holds
( A,B,C are_independent_respect_to P iff ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) )
let A, B, C be Event of Sigma; :: thesis: ( A,B,C are_independent_respect_to P iff ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) )
thus ( A,B,C are_independent_respect_to P implies ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) ) :: thesis: ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P implies A,B,C are_independent_respect_to P )

proof

assume A1: A,B,C are_independent_respect_to P ; :: thesis: ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P )
then A2: P . (B /\ C) = (P . B) * (P . C) by Def6;
( P . (A /\ B) = (P . A) * (P . B) & P . (A /\ C) = (P . A) * (P . C) ) by A1, Def6;
hence ( P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) & A,B are_independent_respect_to P & B,C are_independent_respect_to P & A,C are_independent_respect_to P ) by A1, A2, Def5, Def6; :: thesis: verum

end;

assume that
A3: P . ((A /\ B) /\ C) = ((P . A) * (P . B)) * (P . C) and
A4: A,B are_independent_respect_to P and
A5: B,C are_independent_respect_to P and
A6: A,C are_independent_respect_to P ; :: thesis: A,B,C are_independent_respect_to P
A7: P . (B /\ C) = (P . B) * (P . C) by A5, Def5;
( P . (A /\ B) = (P . A) * (P . B) & P . (A /\ C) = (P . A) * (P . C) ) by A4, A6, Def5;
hence A,B,C are_independent_respect_to P by A3, A7, Def6; :: thesis: verum