let G be non empty multMagma ; :: thesis: ( G is commutative iff for a, b being Element of G holds a * b = b * a )
thus ( G is commutative implies for a, b being Element of G holds a * b = b * a ) :: thesis: ( ( for a, b being Element of G holds a * b = b * a ) implies G is commutative )

proof

assume A1: for a, b being Element of G holds H₂(G) . a,b = H₂(G) . b,a ; :: according to BINOP_1:def 13,MONOID_0:def 11 :: thesis: for a, b being Element of G holds a * b = b * a
let a, b be Element of G; :: thesis: a * b = b * a
thus a * b = b * a by A1; :: thesis: verum

end;

assume A2: for a, b being Element of G holds a * b = b * a ; :: thesis: G is commutative
let a be Element of G; :: according to BINOP_1:def 13,MONOID_0:def 11 :: 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 A2; :: thesis: verum