let K be Ring; :: thesis: for V, W being LeftMod of K
for f being Function of V,W holds
( f is Homomorphism of V,W iff ( ( for x, y being Vector of V holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) ) )

let V, W be LeftMod of K; :: thesis: for f being Function of V,W holds
( f is Homomorphism of V,W iff ( ( for x, y being Vector of V holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) ) )

let f be Function of V,W; :: thesis: ( f is Homomorphism of V,W iff ( ( for x, y being Vector of V holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) ) )

set J = id K;
A1: ( ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) implies for a being Scalar of K
for x being Vector of V holds f . (a * x) = ((id K) . a) * (f . x) ) ;
now :: thesis: ( ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = ((id K) . a) * (f . x) ) implies for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) )
assume A2: for a being Scalar of K
for x being Vector of V holds f . (a * x) = ((id K) . a) * (f . x) ; :: thesis: for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x)

let a be Scalar of K; :: thesis: for x being Vector of V holds f . (a * x) = a * (f . x)
let x be Vector of V; :: thesis: f . (a * x) = a * (f . x)
(id K) . a = a ;
hence f . (a * x) = a * (f . x) by A2; :: thesis: verum
end;
hence ( f is Homomorphism of V,W iff ( ( for x, y being Vector of V holds f . (x + y) = (f . x) + (f . y) ) & ( for a being Scalar of K
for x being Vector of V holds f . (a * x) = a * (f . x) ) ) ) by ; :: thesis: verum