Direction from left to right:

We assume

(B.LA)

and show

(B.RC) .

We prove (B.RC) by the deduction rule.

We assume

(B.RC.H)

and show

(B.RC.C) .

From (B.RC.H) and (B.LA) we obtain by modus ponens

(B.LA.C) .

From (B.RC.H) and (B.LA.C) we obtain by modus ponens

(B.LA.C.C) .

Formula (B.RC.C) is true because it is identical to (B.LA.C.C).