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