let V be RealLinearSpace; :: thesis: for a, b being Real

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let a, b be Real; :: thesis: for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let L be Linear_Combination of V; :: thesis: a * (b * L) = (a * b) * L

let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: (a * (b * L)) . v = ((a * b) * L) . v

thus (a * (b * L)) . v = a * ((b * L) . v) by Def11

.= a * (b * (L . v)) by Def11

.= (a * b) * (L . v)

.= ((a * b) * L) . v by Def11 ; :: thesis: verum

for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let a, b be Real; :: thesis: for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

let L be Linear_Combination of V; :: thesis: a * (b * L) = (a * b) * L

let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: (a * (b * L)) . v = ((a * b) * L) . v

thus (a * (b * L)) . v = a * ((b * L) . v) by Def11

.= a * (b * (L . v)) by Def11

.= (a * b) * (L . v)

.= ((a * b) * L) . v by Def11 ; :: thesis: verum