reserve a,b,r,g for Real;

theorem
  for f be PartFunc of REAL,REAL, a be Real st dom f =
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) holds (for s be Real st s
  in right_open_halfline(0) holds (a(#)f)(#)(exp*-s)
is_+infty_ext_Riemann_integrable_on 0) & One-sided_Laplace_transform(a(#)f) = a
  (#)One-sided_Laplace_transform(f)
proof
  let f be PartFunc of REAL,REAL, a be Real such that
A1: dom f = right_closed_halfline(0) and
A2: for s be Real st s in right_open_halfline(0) holds f(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0;
  set Intf = One-sided_Laplace_transform(f);
  set F = a(#)One-sided_Laplace_transform(f);
A3: dom F = dom Intf by VALUED_1:def 5
    .= right_open_halfline(0) by Def12;
A4: for s be Real st s in right_open_halfline(0) holds (a(#)f)(#)(exp*-s)
  is_+infty_ext_Riemann_integrable_on 0 & infty_ext_right_integral((a(#)f)(#)(
  exp*-s),0) = a*infty_ext_right_integral(f(#)(exp*-s),0)
  proof
    let s be Real;
A5: (a(#)f)(#)(exp*-s) = a(#)(f(#)(exp*-s)) by RFUNCT_1:12;
    assume s in right_open_halfline(0);
    then
A6: f(#)(exp*-s) is_+infty_ext_Riemann_integrable_on 0 by A2;
    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 A6,A5,Th9;
  end;
  for s be Real st s in dom F holds F.s = infty_ext_right_integral((a(#)f)
  (#)(exp*-s),0)
  proof
    let s be Real;
    assume
A7: s in dom F;
    then
A8: s in dom Intf by A3,Def12;
    thus infty_ext_right_integral((a(#)f)(#)(exp*-s),0) = a*
    infty_ext_right_integral(f(#)(exp*-s),0) by A4,A3,A7
      .= a*Intf.s by A8,Def12
      .= F.s by VALUED_1:6;
  end;
  hence thesis by A4,A3,Def12;
end;
