
theorem Th45:
for f be PartFunc of REAL,ExtREAL, a be Real st a in dom f
 ex A be Element of Borel_Sets
  st A = {a} & f is A-measurable & f|A is_integrable_on B-Meas &
   Integral(B-Meas,f|A) = 0
proof
    let f be PartFunc of REAL,ExtREAL, a be Real;
    assume
A1:  a in dom f;

    reconsider A = {a} as Element of Borel_Sets by Lm6;
    take A;
    thus A = {a};
A2: now let r be Real;
     per cases;
     suppose a in less_dom(f,r); then
      A /\ less_dom(f,r) = A by ZFMISC_1:46;
      hence A /\ less_dom(f,r) in Borel_Sets;
     end;
     suppose not a in less_dom(f,r); then
      A /\ less_dom(f,r) = {} by XBOOLE_0:def 7,ZFMISC_1:50;
      hence A /\ less_dom(f,r) in Borel_Sets by MEASURE1:7;
     end;
    end;
    hence
A3:  f is A-measurable by MESFUNC1:def 16;

A4: A = dom f /\ A by A1,ZFMISC_1:31,XBOOLE_1:28; then
A5: f|A is A-measurable by A2,MESFUNC1:def 16,MESFUNC5:42;

    reconsider a1=a as R_eal by XXREAL_0:def 1;
A6: A = [.a,a.] & A = [.a1,a1.] by XXREAL_1:17; then
A7: B-Meas.A = diameter A by MEASUR12:71 .= 0 by A6,MEASURE5:11;

A8: dom(f|A) = A by A4,RELAT_1:61; then
A9: dom(max+(f|A)) = A & dom(max-(f|A)) = A by MESFUNC2:def 2,def 3;

A10: max+(f|A) is A-measurable & max-(f|A) is A-measurable
      by A5,A8,MESFUN11:10; then
    Integral(B-Meas,max+(f|A)|A) = 0 by A7,A9,MESFUNC5:94; then
A11: integral+(B-Meas,max+(f|A)) < +infty by A9,A10,MESFUNC5:88,MESFUN11:5;

    Integral(B-Meas,max-(f|A)|A) = 0 by A7,A9,A10,MESFUNC5:94; then
    integral+(B-Meas,max-(f|A)) = 0 by A9,A10,MESFUNC5:88,MESFUN11:5;
    hence f|A is_integrable_on B-Meas by A8,A4,A11,A3,MESFUNC5:42,def 17;

    Integral(B-Meas,f|A|A) = 0 by A4,A8,A7,A3,MESFUNC5:42,94;
    hence Integral(B-Meas,f|A) = 0;
end;
