
theorem sd:
for X being non empty set
for R being Ring
for L being scalar-distributive non empty ModuleStr over R
for a,b being Element of R
for f being Function of X,L holds (a + b) '*' f = (a '*' f) '+' (b '*' f)
proof
let X be non empty set, F be Ring;
let L be scalar-distributive non empty ModuleStr over F;
let a,b be Element of F, f be Function of X,L;
now let o be object;
  assume o in X;
  then reconsider x = o as Element of X;
  thus ((a + b) '*' f).o
     = (a + b) * f.x by defmu
    .= a * f.x + b * f.x by VECTSP_1:def 15
    .= (a '*' f).x + (b * f.x) by defmu
    .= (a '*' f).x + (b '*' f).x by defmu
    .= ((a '*' f) '+' (b '*' f)).o by defp;
  end;
hence thesis by FUNCT_2:12;
end;
