
theorem Th72:
for f be PartFunc of REAL,REAL, a,b be Real st [.a,b.[ c= dom f &
 max+f is_right_ext_Riemann_integrable_on a,b &
 max-f is_right_ext_Riemann_integrable_on a,b
 holds f is_right_ext_Riemann_integrable_on a,b &
     right_improper_integral(f,a,b)
    = right_improper_integral(max+f,a,b) - right_improper_integral(max-f,a,b)
proof
    let f be PartFunc of REAL,REAL, a,b be Real;
    assume that
A1:  [.a,b.[ c= dom f and
A2:  max+f is_right_ext_Riemann_integrable_on a,b and
A3:  max-f is_right_ext_Riemann_integrable_on a,b;

A4: max+f is_right_improper_integrable_on a,b by A2,INTEGR24:33;
A5: max-f is_right_improper_integrable_on a,b by A3,INTEGR24:33;

    consider I1 be PartFunc of REAL,REAL such that
A6:  dom I1 = [.a,b.[ and
A7:  for x be Real st x in dom I1 holds I1.x = integral(max+f,a,x) and
A8:  I1 is_left_convergent_in b by A2,INTEGR10:def 1;

    consider I2 be PartFunc of REAL,REAL such that
A9:  dom I2 = [.a,b.[ and
A10:  for x be Real st x in dom I2 holds I2.x = integral(max-f,a,x) and
A11:  I2 is_left_convergent_in b by A3,INTEGR10:def 1;

A12:f = max+f - max-f by RFUNCT_3:34;

A13:for d be Real st a <= d & d < b holds
     f is_integrable_on ['a,d'] & f|['a,d'] is bounded
    proof
     let d be Real;
     assume A14: a <= d & d < b; then
A15: max+f is_integrable_on ['a,d'] & max+f|['a,d'] is bounded &
     max-f is_integrable_on ['a,d'] & max-f|['a,d'] is bounded
       by A2,A3,INTEGR10:def 1;
     ['a,d'] = [.a,d.] by A14,INTEGRA5:def 3; then
     ['a,d'] c= [.a,b.[ by A14,XXREAL_1:43; then
     ['a,d'] c= dom f by A1; then
     ['a,d'] c= dom(max+f) & ['a,d'] c= dom(max-f) by RFUNCT_3:def 10,def 11;
     hence f is_integrable_on ['a,d'] by A12,A15,INTEGRA6:11;
     f|(['a,d'] /\ ['a,d']) is bounded by A12,A15,RFUNCT_1:84;
     hence f|['a,d'] is bounded;
    end;

A16:dom(I1-I2) = [.a,b.[ /\ [.a,b.[ by A6,A9,VALUED_1:12;

A17:for x be Real st x in dom(I1-I2) holds (I1-I2).x = integral(f,a,x)
    proof
     let x be Real;
     assume
A18:  x in dom(I1-I2); then
     (I1-I2).x = I1.x - I2.x by VALUED_1:13; then
     (I1-I2).x = integral(max+f,a,x) - I2.x by A18,A6,A7,A16; then
A19: (I1-I2).x = integral(max+f,a,x) - integral(max-f,a,x) by A18,A9,A10,A16;

A20: a <= x & x < b by A18,A16,XXREAL_1:3; then
     ['a,x'] = [.a,x.] by INTEGRA5:def 3; then
     ['a,x'] c= [.a,b.[ by A20,XXREAL_1:43; then
     ['a,x'] c= dom f by A1; then
A21: ['a,x'] c= dom(max+f) & ['a,x'] c= dom(max-f) by RFUNCT_3:def 10,def 11;
     max+f is_integrable_on ['a,x'] & max+f|['a,x'] is bounded &
     max-f is_integrable_on ['a,x'] & max-f|['a,x'] is bounded
       by A20,A2,A3,INTEGR10:def 1;
     hence (I1-I2).x = integral(f,a,x) by A19,A20,A21,A12,INTEGRA6:12;
    end;

A22:for r be Real st r<b ex g be Real st r<g & g<b & g in dom(I1-I2)
      by A16,A6,A8,LIMFUNC2:7;
    hence f is_right_ext_Riemann_integrable_on a,b
      by A13,A16,A17,A8,A11,LIMFUNC2:46,INTEGR10:def 1;

A23:I1-I2 is_left_convergent_in b
  & lim_left(I1-I2,b) = lim_left(I1,b) - lim_left(I2,b)
      by A8,A11,A22,LIMFUNC2:46; then
A24:f is_right_improper_integrable_on a,b
      by A13,A16,A17,INTEGR10:def 1,INTEGR24:33;
A25:lim_left(I1,b) = right_improper_integral(max+f,a,b)
  & lim_left(I2,b) = right_improper_integral(max-f,a,b)
      by A4,A5,A6,A7,A8,A9,A10,A11,INTEGR24:def 4; then
A26: -lim_left(I2,b) = -right_improper_integral(max-f,a,b) by XXREAL_3:def 3;

    right_improper_integral(max+f,a,b) - right_improper_integral(max-f,a,b)
     = right_improper_integral(max+f,a,b) + -right_improper_integral(max-f,a,b)
      by XXREAL_3:def 4
    .= lim_left(I1,b) + -lim_left(I2,b) by A25,A26,XXREAL_3:def 2;
    hence right_improper_integral(f,a,b)
      = right_improper_integral(max+f,a,b) - right_improper_integral(max-f,a,b)
       by A24,A16,A17,A23,INTEGR24:def 4;
end;
