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

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

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

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

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

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

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

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

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

.= ((a * L) + (b * L)) . v by Def10 ; :: thesis: verum

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

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

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

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

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

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

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

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

.= ((a * L) + (b * L)) . v by Def10 ; :: thesis: verum