reserve a,b,r,g for Real;

theorem
  for f,g be PartFunc of REAL,REAL, b be Real st
  left_closed_halfline(b) c= dom f & left_closed_halfline(b) c= dom g & f
is_-infty_ext_Riemann_integrable_on b & g is_-infty_ext_Riemann_integrable_on b
holds f + g is_-infty_ext_Riemann_integrable_on b & infty_ext_left_integral(f +
  g,b) = infty_ext_left_integral(f,b) + infty_ext_left_integral(g,b)
proof
  let f,g be PartFunc of REAL,REAL, b be Real;
  assume that
A1: left_closed_halfline(b) c= dom f & left_closed_halfline(b) c= dom g and
A2: f is_-infty_ext_Riemann_integrable_on b and
A3: g is_-infty_ext_Riemann_integrable_on b;
  consider Intg be PartFunc of REAL,REAL such that
A4: dom Intg = left_closed_halfline(b) and
A5: for x be Real st x in dom Intg holds Intg.x = integral(g,x,b) and
A6: Intg is convergent_in-infty and
A7: infty_ext_left_integral(g,b) = lim_in-infty Intg by A3,Def8;
  consider Intf be PartFunc of REAL,REAL such that
A8: dom Intf = left_closed_halfline(b) and
A9: for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A10: Intf is convergent_in-infty and
A11: infty_ext_left_integral(f,b) = lim_in-infty Intf by A2,Def8;
  set Intfg = Intf + Intg;
A12: dom Intfg = left_closed_halfline(b) & for x be Real st x in dom Intfg
  holds Intfg.x = integral(f + g,x,b)
  proof
    thus
A13: dom Intfg = dom Intf /\ dom Intg by VALUED_1:def 1
      .= left_closed_halfline(b) by A8,A4;
    let x be Real;
    assume
A14: x in dom Intfg;
    then
A15: x <= b by A13,XXREAL_1:234;
    then
A16: f is_integrable_on [' x,b '] & f|[' x,b '] is bounded by A2;
A17: [.x,b.] c= left_closed_halfline(b) by XXREAL_1:265;
    [' x,b '] = [.x,b.] by A15,INTEGRA5:def 3;
    then
A18: [' x,b '] c= dom f & [' x,b '] c= dom g by A1,A17;
A19: g is_integrable_on [' x,b '] & g|[' x,b '] is bounded by A3,A15;
    thus Intfg.x = Intf.x + Intg.x by A14,VALUED_1:def 1
      .= integral(f,x,b) + Intg.x by A8,A9,A13,A14
      .= integral(f,x,b) + integral(g,x,b) by A4,A5,A13,A14
      .= integral(f + g,x,b) by A15,A18,A16,A19,INTEGRA6:12;
  end;
A20: for r ex g st g < r & g in dom(Intf + Intg)
  proof
    let r be Real;
    per cases;
    suppose
A21:  b < r;
      reconsider g = b as Real;
      take g;
      thus thesis by A12,A21,XXREAL_1:234;
    end;
    suppose
A22:  not b < r;
      reconsider g = r - 1 as Real;
      take g;
A23:  r - 1 < r - 0 by XREAL_1:15;
      then g <= b by A22,XXREAL_0:2;
      hence thesis by A12,A23,XXREAL_1:234;
    end;
  end;
  then
A24: Intfg is convergent_in-infty by A10,A6,LIMFUNC1:91;
  for a be Real st a <= b holds f + g is_integrable_on [' a,b '] & (f+g)|
  [' a,b '] is bounded
  proof
    let a be Real;
A25: [.a,b.] c= left_closed_halfline(b) by XXREAL_1:265;
    assume
A26: a <= b;
    then
A27: f is_integrable_on [' a,b '] & g is_integrable_on [' a,b '] by A2,A3;
    [' a,b '] = [.a,b.] by A26,INTEGRA5:def 3;
    then
A28: [' a,b '] c= dom f & [' a,b '] c= dom g by A1,A25;
A29: f|[' a,b '] is bounded & g|[' a,b '] is bounded by A2,A3,A26;
    then (f + g)|([' a,b '] /\ [' a,b ']) is bounded by RFUNCT_1:83;
    hence thesis by A28,A27,A29,INTEGRA6:11;
  end;
  hence
A30: (f + g) is_-infty_ext_Riemann_integrable_on b by A12,A24;
  lim_in-infty (Intfg) = infty_ext_left_integral(f,b) +
  infty_ext_left_integral(g,b) by A10,A11,A6,A7,A20,LIMFUNC1:91;
  hence thesis by A12,A24,A30,Def8;
end;
