
theorem vd:
for X being non empty set
for L being right-distributive non empty doubleLoopStr
for a being Element of L
for f,g being Function of X,L holds a '*' (f '+' g) = (a '*' f) '+' (a '*' g)
proof
let X be non empty set, L be right-distributive non empty doubleLoopStr,
    a be Element of L, 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 2
    .= 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;
