
theorem Th15:
  for K be right-distributive non empty doubleLoopStr for V be
  non empty ModuleStr over K for r be Element of K for f,g be Functional of V
  holds r*(f+g) = r*f+r*g
proof
  let K be right-distributive non empty doubleLoopStr;
  let V be non empty ModuleStr over K;
  let r be Element of K;
  let f,g be Functional of V;
  now
    let x be Element of V;
    thus (r*(f+g)).x = r*(f+g).x by Def6
      .= r*(f.x+g.x) by Def3
      .= r*f.x+r*g.x by VECTSP_1:def 2
      .= (r*f).x+r*g.x by Def6
      .= (r*f).x+(r*g).x by Def6
      .= (r*f+r*g).x by Def3;
  end;
  hence thesis by FUNCT_2:63;
end;
