
theorem Th16:
  for K be left-distributive non empty doubleLoopStr for V be
  non empty ModuleStr over K for r,s be Element of K for f be Functional of V
  holds (r+s)*f = r*f+s*f
proof
  let K be left-distributive non empty doubleLoopStr;
  let V be non empty ModuleStr over K;
  let r,s be Element of K;
  let f be Functional of V;
  now
    let x be Element of V;
    thus ((r+s)*f).x = (r+s)*f.x by Def6
      .= r*f.x+s*f.x by VECTSP_1:def 3
      .= (r*f).x+s*f.x by Def6
      .= (r*f).x+(s*f).x by Def6
      .= (r*f+s*f).x by Def3;
  end;
  hence thesis by FUNCT_2:63;
end;
