thus ( G is commutative implies the multF of G is commutative ) :: thesis: ( the multF of G is commutative implies G is commutative )

proof

assume A1: for a, b being Element of G holds a * b = b * a ; :: according to GROUP_1:def 12 :: thesis: the multF of G is commutative
let a be Element of G; :: according to BINOP_1:def 2 :: thesis: for b₁ being Element of the carrier of G holds the multF of G . (a,b₁) = the multF of G . (b₁,a)
let b be Element of G; :: thesis: the multF of G . (a,b) = the multF of G . (b,a)
( H₂(G) . (a,b) = a * b & H₂(G) . (b,a) = b * a ) ;
hence the multF of G . (a,b) = the multF of G . (b,a) by A1; :: thesis: verum

end;

assume A2: for a, b being Element of G holds H₂(G) . (a,b) = H₂(G) . (b,a) ; :: according to BINOP_1:def 2 :: thesis: G is commutative
let a be Element of G; :: according to GROUP_1:def 12 :: thesis: for b₁ being Element of the carrier of G holds a [*] b₁ = b₁ [*] a
let b be Element of G; :: thesis: a [*] b = b [*] a
thus a [*] b = b [*] a by A2; :: thesis: verum