let V be Z_Module; :: thesis: for L1, L2 being Z_Linear_Combination of V
for a being Integer holds a * (L1 + L2) = (a * L1) + (a * L2)
let L1, L2 be Z_Linear_Combination of V; :: thesis: for a being Integer holds a * (L1 + L2) = (a * L1) + (a * L2)
let a be Integer; :: thesis: a * (L1 + L2) = (a * L1) + (a * L2)
let v be VECTOR of V; :: according to ZMODUL02:def 24 :: thesis: (a * (L1 + L2)) . v = ((a * L1) + (a * L2)) . v
thus (a * (L1 + L2)) . v = a * ((L1 + L2) . v) by Def26
.= a * ((L1 . v) + (L2 . v)) by Def25
.= (a * (L1 . v)) + (a * (L2 . v))
.= ((a * L1) . v) + (a * (L2 . v)) by Def26
.= ((a * L1) . v) + ((a * L2) . v) by Def26
.= ((a * L1) + (a * L2)) . v by Def25 ; :: thesis: verum