
theorem
  for V being non empty ModuleStr over INT.Ring, r being Element of F_Real,
  f, g being FrFunctional of V holds r*(f+g) = r*f+r*g
  proof
    let V be non empty ModuleStr over INT.Ring;
    let r be Element of F_Real;
    let f, g be FrFunctional of V;
    now
      let x be Element of V;
      thus (r*(f+g)).x = r*(f+g).x by HDef6
      .= r*(f.x+g.x) by HDef3
      .= r*f.x+r*g.x
      .= (r*f).x+r*g.x by HDef6
      .= (r*f).x+(r*g).x by HDef6
      .= (r*f+r*g).x by HDef3;
    end;
    hence thesis by FUNCT_2:63;
  end;
