let a, b be Element of K; :: according to VECTSP_1:def 15 :: thesis: for b_{1} being Element of the carrier of [:G,F:] holds (a * b) * b_{1} = a * (b * b_{1})

let x be Vector of [:G,F:]; :: thesis: (a * b) * x = a * (b * x)

consider x1 being Vector of G, x2 being Vector of F such that

A1: x = [x1,x2] by SUBSET_1:43;

A2: ( (a * b) * x1 = a * (b * x1) & (a * b) * x2 = a * (b * x2) ) by VECTSP_1:def 16;

thus (a * b) * x = [((a * b) * x1),((a * b) * x2)] by A1, YDef2

.= (prod_MLT (G,F)) . (a,[(b * x1),(b * x2)]) by A2, YDef2

.= a * (b * x) by A1, YDef2 ; :: thesis: verum

let x be Vector of [:G,F:]; :: thesis: (a * b) * x = a * (b * x)

consider x1 being Vector of G, x2 being Vector of F such that

A1: x = [x1,x2] by SUBSET_1:43;

A2: ( (a * b) * x1 = a * (b * x1) & (a * b) * x2 = a * (b * x2) ) by VECTSP_1:def 16;

thus (a * b) * x = [((a * b) * x1),((a * b) * x2)] by A1, YDef2

.= (prod_MLT (G,F)) . (a,[(b * x1),(b * x2)]) by A2, YDef2

.= a * (b * x) by A1, YDef2 ; :: thesis: verum