reserve a,b,r,g for Real;

theorem
  for f be PartFunc of REAL,REAL, a,b be Real st a < b & [' a,b '] c=
dom f & f is_left_ext_Riemann_integrable_on a,b holds for r be Real holds r(#)f
  is_left_ext_Riemann_integrable_on a,b & ext_left_integral(r(#)f,a,b) = r*
  ext_left_integral(f,a,b)
proof
  let f be PartFunc of REAL,REAL, a,b be Real such that
A1: a < b and
A2: [' a,b '] c= dom f and
A3: f is_left_ext_Riemann_integrable_on a,b;
  for r be Real holds r(#)f is_left_ext_Riemann_integrable_on a,b &
  ext_left_integral(r(#)f,a,b) = r*ext_left_integral(f,a,b)
  proof
    let r be Real;
    consider Intf be PartFunc of REAL,REAL such that
A4: dom Intf = ].a,b.] and
A5: for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A6: Intf is_right_convergent_in a and
A7: ext_left_integral(f,a,b) = lim_right(Intf,a) by A3,Def4;
    set Intfg = r(#)Intf;
A8: Intfg is_right_convergent_in a by A6,LIMFUNC2:52;
A9: dom Intfg = ].a,b.] &
   for x be Real st x in dom Intfg holds Intfg.x =
    integral(r(#)f,x,b)
    proof
A10:  [' a,b '] = [.a,b.] by A1,INTEGRA5:def 3;
      thus
A11:  dom Intfg = ].a,b.] by A4,VALUED_1:def 5;
      let x be Real;
      assume
A12:  x in dom Intfg;
      then
A13:  a < x by A11,XXREAL_1:2;
      then
A14:  [.x,b.] c= [.a,b.] by XXREAL_1:34;
A15:  x <= b by A11,A12,XXREAL_1:2;
      then
A16:  [' x,b '] = [.x,b.] by INTEGRA5:def 3;
A17:  f is_integrable_on [' x,b '] & f|[' x,b '] is bounded by A3,A13,A15;
      thus Intfg.x = r*Intf.x by A12,VALUED_1:def 5
        .= r*integral(f,x,b) by A4,A5,A11,A12
        .= integral(r(#)f,x,b) by A2,A15,A16,A10,A14,A17,INTEGRA6:10,XBOOLE_1:1
;
    end;
    for d be Real st a < d & d <= b
holds r(#)f is_integrable_on [' d,b ']
    & (r(#)f)|[' d,b '] is bounded
    proof
      let d be Real;
      assume
A18:  a < d & d <= b;
      then
A19:  [' d,b '] = [.d,b.] & [.d,b.] c= [.a,b.] by INTEGRA5:def 3,XXREAL_1:34;
A20:  f is_integrable_on [' d,b '] & f|[' d,b '] is bounded by A3,A18;
      [' a,b '] = [.a,b.] by A1,INTEGRA5:def 3;
      then [' d,b '] c= dom f by A2,A19;
      hence thesis by A20,INTEGRA6:9,RFUNCT_1:80;
    end;
    hence
A21: r(#)f is_left_ext_Riemann_integrable_on a,b by A9,A8;
    lim_right (Intfg,a) = r*ext_left_integral(f,a,b) by A6,A7,LIMFUNC2:52;
    hence thesis by A9,A8,A21,Def4;
  end;
  hence thesis;
end;
