reserve a,b,r,g for Real;

theorem Th8:
  for f,g be PartFunc of REAL,REAL, a be Real st
  right_closed_halfline(a) c= dom f & right_closed_halfline(a) c= dom g & f
is_+infty_ext_Riemann_integrable_on a & g is_+infty_ext_Riemann_integrable_on a
holds f + g is_+infty_ext_Riemann_integrable_on a & infty_ext_right_integral(f
  + g,a) = infty_ext_right_integral(f,a) + infty_ext_right_integral(g,a)
proof
  let f,g be PartFunc of REAL,REAL, a be Real;
  assume that
A1: right_closed_halfline(a) c= dom f & right_closed_halfline(a) c= dom g and
A2: f is_+infty_ext_Riemann_integrable_on a and
A3: g is_+infty_ext_Riemann_integrable_on a;
  consider Intg be PartFunc of REAL,REAL such that
A4: dom Intg = right_closed_halfline(a) and
A5: for x be Real st x in dom Intg holds Intg.x = integral(g,a,x) and
A6: Intg is convergent_in+infty and
A7: infty_ext_right_integral(g,a) = lim_in+infty Intg by A3,Def7;
  consider Intf be PartFunc of REAL,REAL such that
A8: dom Intf = right_closed_halfline(a) and
A9: for x be Real st x in dom Intf holds Intf.x = integral(f,a,x) and
A10: Intf is convergent_in+infty and
A11: infty_ext_right_integral(f,a) = lim_in+infty Intf by A2,Def7;
  set Intfg = Intf + Intg;
A12: dom Intfg = right_closed_halfline(a) & for x be Real st x in dom Intfg
  holds Intfg.x = integral(f + g,a,x)
  proof
    thus
A13: dom Intfg = dom Intf /\ dom Intg by VALUED_1:def 1
      .= right_closed_halfline(a) by A8,A4;
    let x be Real;
    assume
A14: x in dom Intfg;
    then
A15: a <= x by A13,XXREAL_1:236;
    then
A16: f is_integrable_on [' a,x '] & f|[' a,x '] is bounded by A2;
A17: [.a,x.] c= right_closed_halfline(a) by XXREAL_1:251;
    [' a,x '] = [.a,x.] by A15,INTEGRA5:def 3;
    then
A18: [' a,x '] c= dom f & [' a,x '] c= dom g by A1,A17;
A19: g is_integrable_on [' a,x '] & g|[' a,x '] is bounded by A3,A15;
    thus Intfg.x = Intf.x + Intg.x by A14,VALUED_1:def 1
      .= integral(f,a,x) + Intg.x by A8,A9,A13,A14
      .= integral(f,a,x) + integral(g,a,x) by A4,A5,A13,A14
      .= integral(f + g,a,x) by A15,A18,A16,A19,INTEGRA6:12;
  end;
A20: for r ex g st r < g & g in dom(Intf + Intg)
  proof
    let r be Real;
    per cases;
    suppose
A21:  r < a;
      reconsider g = a as Real;
      take g;
      thus thesis by A12,A21,XXREAL_1:236;
    end;
    suppose
A22:  not r < a;
      reconsider g = r + 1 as Real;
      take g;
A23:  r + 0 < r + 1 by XREAL_1:8;
      then a <= g by A22,XXREAL_0:2;
      hence thesis by A12,A23,XXREAL_1:236;
    end;
  end;
  then
A24: Intfg is convergent_in+infty by A10,A6,LIMFUNC1:82;
  for b be Real st a <= b holds f + g is_integrable_on [' a,b '] & (f+g)|
  [' a,b '] is bounded
  proof
    let b be Real;
A25: [.a,b.] c= right_closed_halfline(a) by XXREAL_1:251;
    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 a by A12,A24;
  lim_in+infty (Intfg) = infty_ext_right_integral(f,a) +
  infty_ext_right_integral(g,a) by A10,A11,A6,A7,A20,LIMFUNC1:82;
  hence thesis by A12,A24,A30,Def7;
end;
