
theorem Th81:
for f be PartFunc of REAL,REAL, a,b be Real, A be non empty Subset of REAL
 st ].a,b.] c= dom f & A = ].a,b.] & f is_left_improper_integrable_on a,b
 & abs f is_left_ext_Riemann_integrable_on a,b
 holds f|A is_integrable_on L-Meas
     & left_improper_integral(f,a,b) = Integral(L-Meas,f|A)
proof
    let f be PartFunc of REAL,REAL, a,b be Real, A be non empty Subset of REAL;
    assume that
A1:  ].a,b.] c= dom f and
A2:  A = ].a,b.] and
A3:  f is_left_improper_integrable_on a,b and
A4:  abs f is_left_ext_Riemann_integrable_on a,b;

A5:  dom max+f = dom f by RFUNCT_3:def 10;

A6:  a < b by A2,XXREAL_1:26; then
A7:  f is_left_ext_Riemann_integrable_on a,b by A1,A3,A4,Th58; then
A8:  max+f is_left_improper_integrable_on a,b
       by A1,A4,A6,Th63,INTEGR24:32;

A9:  max+f is_left_ext_Riemann_integrable_on a,b
       by A1,A4,A2,A7,Th63,XXREAL_1:26; then
A10: abs(max+f) is_left_ext_Riemann_integrable_on a,b
       by MESFUNC6:61,LPSPACE2:14;

A11: (max+f)|A is nonnegative by MESFUNC6:61,55; then
     (max+f)|A is_integrable_on L-Meas by A1,A5,A2,A8,A10,Th75; then
A12: max+(f|A) is_integrable_on L-Meas by MESFUNC6:66;

     max+(R_EAL(f|A)) = max+(f|A) by MESFUNC6:30; then
A13: max+(R_EAL(f|A)) = R_EAL(max+(f|A)) by MESFUNC5:def 7; then
A14: max+(R_EAL(f|A)) is_integrable_on L-Meas by A12,MESFUNC6:def 4;

A15: dom max-f = dom f by RFUNCT_3:def 11;

A16: max-f is_left_improper_integrable_on a,b
       by A1,A4,A6,A7,Th67,INTEGR24:32;

A17: max-f is_left_ext_Riemann_integrable_on a,b
      by A1,A4,A2,A7,Th67,XXREAL_1:26; then
A18: abs(max-f) is_left_ext_Riemann_integrable_on a,b
       by LPSPACE2:14,MESFUNC6:61;

A19: (max-f)|A is nonnegative by MESFUNC6:61,55; then
     (max-f)|A is_integrable_on L-Meas by A1,A15,A2,A16,A18,Th75; then
A20: max-(f|A) is_integrable_on L-Meas by MESFUNC6:66;

     max-(R_EAL(f|A)) = max-(f|A) by MESFUNC6:30; then
A21: max-(R_EAL(f|A)) = R_EAL(max-(f|A)) by MESFUNC5:def 7; then
     max-(R_EAL(f|A)) is_integrable_on L-Meas by A20,MESFUNC6:def 4;
     hence f|A is_integrable_on L-Meas by A14,MESFUN13:18,MESFUNC6:def 4;

     reconsider A1=A as Element of L-Field by A2,MEASUR12:72,MEASUR12:75;

     R_EAL(f|A) is_integrable_on L-Meas
       by A21,A14,A20,MESFUNC6:def 4,MESFUN13:18; then
     consider E be Element of L-Field such that
A22:   E = dom(R_EAL(f|A)) & R_EAL(f|A) is E-measurable by MESFUNC5:def 17;

A23:  left_improper_integral(f,a,b)
      = left_improper_integral(max+f,a,b) - left_improper_integral(max-f,a,b)
       by A1,A9,A17,Th71;
A24:  left_improper_integral(max+f,a,b)
      = Integral(L-Meas,(max+f)|A) by A1,A5,A2,A8,A10,A11,Th75
     .= Integral(L-Meas,max+(f|A)) by MESFUNC6:66
     .= Integral(L-Meas,max+(R_EAL(f|A))) by A13,MESFUNC6:def 3;

     left_improper_integral(max-f,a,b)
      = Integral(L-Meas,(max-f)|A) by A1,A15,A2,A16,A18,A19,Th75
     .= Integral(L-Meas,max-(f|A)) by MESFUNC6:66
     .= Integral(L-Meas,max-(R_EAL(f|A))) by A21,MESFUNC6:def 3; then
     left_improper_integral(f,a,b) = Integral(L-Meas,R_EAL(f|A))
       by A22,A23,A24,MESFUN11:54;
     hence left_improper_integral(f,a,b) = Integral(L-Meas,f|A)
       by MESFUNC6:def 3;
end;
