let a, b be Real; :: thesis: for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
for v being Element of V holds (a - b) * v = (a * v) - (b * v)
let V be non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct ; :: thesis: for v being Element of V holds (a - b) * v = (a * v) - (b * v)
let v be Element of V; :: thesis: (a - b) * v = (a * v) - (b * v)
thus (a - b) * v = (a + (- b)) * v
.= (a * v) + ((- b) * v) by Def6
.= (a * v) + (b * (- v)) by Th24
.= (a * v) - (b * v) by Th25 ; :: thesis: verum