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