let V be RealLinearSpace; :: thesis: for a being Real
for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)
let a be Real; :: thesis: for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)
let L1, L2 be Linear_Combination of V; :: thesis: a * (L1 + L2) = (a * L1) + (a * L2)
let v be VECTOR of V; :: according to RLVECT_2:def 11 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v
thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def13
.= a * ((L1 . v) + (L2 . v)) by Def12
.= (a * (L1 . v)) + (a * (L2 . v))
.= ((a * L1) . v) + (a * (L2 . v)) by Def13
.= ((a * L1) . v) + ((a * L2) . v) by Def13
.= ((a * L1) + (a * L2)) . v by Def12 ; :: thesis: verum