let A be non empty set ; :: thesis: for o, o9 being BinOp of A st o9 is commutative holds
( o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o )
let o, o9 be BinOp of A; :: thesis: ( o9 is commutative implies ( o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o ) )
( o9 is_left_distributive_wrt o iff for a, b, c being Element of A holds o9 . (a,(o . (b,c))) = o . ((o9 . (a,b)),(o9 . (a,c))) ) by Def9;
hence ( o9 is commutative implies ( o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o ) ) by Th24; :: thesis: verum