
theorem Th30:
  for f be PartFunc of REAL,REAL, a,b be Real st ].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.] c= dom f and
A2: 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
A3: dom Intf = ].a,b.] and
A4: for x be Real st x in dom Intf holds Intf.x = integral(f,x,b) and
A5: Intf is_right_convergent_in a and
A6: ext_left_integral(f,a,b) = lim_right(Intf,a) by A2,INTEGR10:def 4;
    set Intfg = r(#)Intf;
A7: Intfg is_right_convergent_in a by A5,LIMFUNC2:52;
A8: dom Intfg = ].a,b.] &
   for x be Real st x in dom Intfg holds Intfg.x =
    integral(r(#)f,x,b)
    proof
      thus
A9:  dom Intfg = ].a,b.] by A3,VALUED_1:def 5;
      let x be Real;
      assume
A10:  x in dom Intfg;
      then
A11:  a < x by A9,XXREAL_1:2;
      then
A12:  [.x,b.] c= ].a,b.] by XXREAL_1:39;
A13:  x <= b by A9,A10,XXREAL_1:2;
      then
A14:  [' x,b '] = [.x,b.] by INTEGRA5:def 3;
A15:  f is_integrable_on [' x,b '] & f|[' x,b '] is bounded
by A2,A11,A13,INTEGR10:def 2;
      thus Intfg.x = r*Intf.x by A10,VALUED_1:def 5
        .= r*integral(f,x,b) by A3,A4,A9,A10
        .= integral(r(#)f,x,b) by A1,A13,A14,A12,A15,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
A16:  a < d & d <= b;
      then
A17:  [' d,b '] = [.d,b.] & [.d,b.] c= ].a,b.] by INTEGRA5:def 3,XXREAL_1:39;
A18:  f is_integrable_on [' d,b '] & f|[' d,b '] is bounded
by A2,A16,INTEGR10:def 2;
      [' d,b '] c= dom f by A1,A17;
      hence thesis by A18,INTEGRA6:9,RFUNCT_1:80;
    end;
    hence
A19: r(#)f is_left_ext_Riemann_integrable_on a,b
      by A8,A5,LIMFUNC2:52,INTEGR10:def 2;
    lim_right (Intfg,a) = r*ext_left_integral(f,a,b) by A5,A6,LIMFUNC2:52;
    hence thesis by A8,A7,A19,INTEGR10:def 4;
  end;
  hence thesis;
end;
