let a0, b0 be Real; :: according to RLVECT_1:def 7 :: thesis: for b₁ being Element of the carrier of [:G,F:] holds (a0 * b0) * b₁ = a0 * (b0 * b₁)
let x be VECTOR of [:G,F:]; :: thesis: (a0 * b0) * x = a0 * (b0 * x)
reconsider a = a0, b = b0 as Element of REAL by XREAL_0:def 1;
consider x1 being Point of G, x2 being Point of F such that
A1: x = [x1,x2] by Lm1;
A2: ( (a * b) * x1 = a0 * (b0 * x1) & (a * b) * x2 = a0 * (b0 * x2) ) by RLVECT_1:def 7;
thus (a0 * b0) * x = [((a * b) * x1),((a * b) * x2)] by A1, Def2
.= (prod_MLT (G,F)) . (a,[(b * x1),(b * x2)]) by A2, Def2
.= a0 * (b0 * x) by A1, Def2 ; :: thesis: verum