let a, b be Integer; :: thesis: for V being Z_Module
for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)
let V be Z_Module; :: thesis: for v being VECTOR of V holds (a - b) * v = (a * v) - (b * v)
let v be VECTOR of V; :: thesis: (a - b) * v = (a * v) - (b * v)
thus (a - b) * v = (a + (- b)) * v
.= (a * v) + ((- b) * v) by Def3
.= (a * v) + (b * (- v)) by Th5
.= (a * v) - (b * v) by Th6 ; :: thesis: verum