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 A = ((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 A = ((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
; prob A = ((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))
A5:
B2 \ B1 = E \ B1
by A3, XBOOLE_1:40;
then
0 < prob (B1 `)
by A2, A4, XBOOLE_1:83;
then
0 < 1 - (prob B1)
by Th22;
then A6:
1 - (1 - (prob B1)) < 1
by XREAL_1:44;
B2 = B1 `
by A4, A5, XBOOLE_1:83;
hence
prob A = ((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))
by A1, A6, Th49; verum