let E be non empty finite set ; for A, B1, B2 being Event of E st 0 < prob B1 & 0 < prob B2 & B1 \/ B2 = E & B1 misses B2 holds
prob (B1,A) = ((prob (A,B1)) * (prob B1)) / (((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2)))
let A, B1, B2 be Event of E; ( 0 < prob B1 & 0 < prob B2 & B1 \/ B2 = E & B1 misses B2 implies prob (B1,A) = ((prob (A,B1)) * (prob B1)) / (((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) )
assume that
A1:
0 < prob B1
and
A2:
( 0 < prob B2 & B1 \/ B2 = E & B1 misses B2 )
; prob (B1,A) = ((prob (A,B1)) * (prob B1)) / (((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2)))
prob A = ((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))
by A1, A2, Th50;
hence
prob (B1,A) = ((prob (A,B1)) * (prob B1)) / (((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2)))
by A1, XCMPLX_1:87; verum