let V be non empty CLSStruct ; :: thesis: for a being Complex
for L1, L2 being C_Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)
let a be Complex; :: thesis: for L1, L2 being C_Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)
let L1, L2 be C_Linear_Combination of V; :: thesis: a * (L1 + L2) = (a * L1) + (a * L2)
let v be VECTOR of V; :: according to CONVEX4:def 7 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v
thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def9
.= a * ((L1 . v) + (L2 . v)) by Def8
.= (a * (L1 . v)) + (a * (L2 . v))
.= ((a * L1) . v) + (a * (L2 . v)) by Def9
.= ((a * L1) . v) + ((a * L2) . v) by Def9
.= ((a * L1) + (a * L2)) . v by Def8 ; :: thesis: verum