let G be Group; :: thesis: ( G is commutative Group iff the multF of G is commutative )
thus ( G is commutative Group implies the multF of G is commutative ) :: thesis: ( the multF of G is commutative implies G is commutative Group )
proof
assume A1: G is commutative Group ; :: 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)
thus the multF of G . (a,b) = a * b
.= b * a by A1, Lm1
.= the multF of G . (b,a) ; :: thesis: verum
end;
assume A2: the multF of G is commutative ; :: thesis: G is commutative Group
G is commutative by A2;
hence G is commutative Group ; :: thesis: verum