reserve a,b,r,g for Real;

theorem
  for f, g be PartFunc of REAL,REAL st dom f = right_closed_halfline(0)
& dom g = right_closed_halfline(0) & (for s be Real st s in right_open_halfline
(0) holds f(#)(exp*-s) is_+infty_ext_Riemann_integrable_on 0) & (for s be Real
  st s in right_open_halfline(0) holds g(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0) holds (for s be Real st s in
  right_open_halfline(0) holds (f + g)(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0) & One-sided_Laplace_transform(f + g) =
  One-sided_Laplace_transform(f) + One-sided_Laplace_transform(g)
proof
  let f, g be PartFunc of REAL,REAL such that
A1: dom f = right_closed_halfline(0) and
A2: dom g = right_closed_halfline(0) and
A3: ( for s be Real st s in right_open_halfline(0) holds f(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0)& for s be Real st s in
right_open_halfline(0) holds g(#)(exp*-s) is_+infty_ext_Riemann_integrable_on 0
  ;
  set Intg = One-sided_Laplace_transform(g);
  set Intf = One-sided_Laplace_transform(f);
  set F = One-sided_Laplace_transform(f) + One-sided_Laplace_transform(g);
A4: dom F = dom Intf /\ dom Intg by VALUED_1:def 1
    .= right_open_halfline(0) /\ dom Intg by Def12
    .= right_open_halfline(0) /\ right_open_halfline(0) by Def12
    .= right_open_halfline(0);
A5: for s be Real st s in right_open_halfline(0) holds (f + g)(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0 & infty_ext_right_integral((f + g)(#)(
  exp*-s),0) = infty_ext_right_integral(f(#)(exp*-s),0) +
  infty_ext_right_integral(g(#)(exp*-s),0)
  proof
    let s be Real;
A6: (f + g)(#)(exp*-s) = f(#)(exp*-s) + g(#)(exp*-s) by RFUNCT_1:10;
A7: dom (g(#)(exp*-s)) = dom g /\ dom (exp*-s) by VALUED_1:def 4
      .= right_closed_halfline(0) /\ REAL by A2,FUNCT_2:def 1
      .= right_closed_halfline(0) by XBOOLE_1:28;
    assume s in right_open_halfline(0);
    then
A8: f(#)(exp*-s) is_+infty_ext_Riemann_integrable_on 0 & g(#)(exp*-s)
    is_+infty_ext_Riemann_integrable_on 0 by A3;
    dom (f(#)(exp*-s)) = dom f /\ dom (exp*-s) by VALUED_1:def 4
      .= right_closed_halfline(0) /\ REAL by A1,FUNCT_2:def 1
      .= right_closed_halfline(0) by XBOOLE_1:28;
    hence thesis by A8,A6,A7,Th8;
  end;
  for s be Real st s in dom F holds F.s = infty_ext_right_integral((f + g
  )(#)(exp*-s),0)
  proof
    let s be Real;
    assume
A9: s in dom F;
    then
A10: s in dom Intf by A4,Def12;
A11: s in dom Intg by A4,A9,Def12;
    thus infty_ext_right_integral((f + g)(#)(exp*-s),0) =
infty_ext_right_integral(f(#)(exp*-s),0) + infty_ext_right_integral(g(#)(exp*-s
    ),0) by A5,A4,A9
      .= Intf.s + infty_ext_right_integral(g(#)(exp*-s),0) by A10,Def12
      .= Intf.s + Intg.s by A11,Def12
      .= F.s by A9,VALUED_1:def 1;
  end;
  hence thesis by A5,A4,Def12;
end;
