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 9 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v

thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def11

.= a * ((L1 . v) + (L2 . v)) by Def10

.= (a * (L1 . v)) + (a * (L2 . v))

.= ((a * L1) . v) + (a * (L2 . v)) by Def11

.= ((a * L1) . v) + ((a * L2) . v) by Def11

.= ((a * L1) + (a * L2)) . v by Def10 ; :: thesis: verum

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 9 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v

thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def11

.= a * ((L1 . v) + (L2 . v)) by Def10

.= (a * (L1 . v)) + (a * (L2 . v))

.= ((a * L1) . v) + (a * (L2 . v)) by Def11

.= ((a * L1) . v) + ((a * L2) . v) by Def11

.= ((a * L1) + (a * L2)) . v by Def10 ; :: thesis: verum