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