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