let a, b be Element of K; :: according to VECTSP_1:def 14 :: thesis: for b₁ being Element of the carrier of [:G,F:] holds (a + b) * b₁ = (a * b₁) + (b * b₁)
let x be Vector of [:G,F:]; :: thesis: (a + b) * x = (a * x) + (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 * x1) + (b * x1) & (a + b) * x2 = (a * x2) + (b * x2) ) by VECTSP_1:def 15;
thus (a + b) * x = [((a + b) * x1),((a + b) * x2)] by A1, YDef2
.= (prod_ADD (G,F)) . ([(a * x1),(a * x2)],[(b * x1),(b * x2)]) by A2, PRVECT_3:def 1
.= (prod_ADD (G,F)) . (((prod_MLT (G,F)) . (a,[x1,x2])),[(b * x1),(b * x2)]) by YDef2
.= (a * x) + (b * x) by A1, YDef2 ; :: thesis: verum