let E be non empty finite set ; :: thesis: 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; :: thesis: ( 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 and
A3: B1 \/ B2 = E and
A4: B1 misses B2 ; :: thesis: 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, A3, A4, Th80;
hence prob B1,A = ((prob A,B1) * (prob B1)) / (((prob A,B1) * (prob B1)) + ((prob A,B2) * (prob B2))) by A1, XCMPLX_1:88; :: thesis: verum