
theorem vd:
for X being non empty set
for R being Ring
for L being vector-distributive non empty ModuleStr over R
for a,b being Element of R
for f,g being Function of X,L holds a '*' (f '+' g) = (a '*' f) '+' (a '*' g)
proof
let X be non empty set, R be Ring,
    L being vector-distributive non empty ModuleStr over R,
    a,b be Element of R, f,g be Function of X,L;
now let o be object;
  assume o in X;
  then reconsider x = o as Element of X;
  thus (a '*' (f '+' g)).o
     = a * ((f'+'g).x) by defmu
    .= a * (f.x + g.x) by defp
    .= a * f.x + a * g.x by VECTSP_1:def 14
    .= a * f.x + (a '*' g).x by defmu
    .= (a '*' f).x + (a '*' g).x by defmu
    .= ((a '*' f) '+' (a '*' g)).o by defp;
  end;
hence thesis by FUNCT_2:12;
end;
