
theorem Th46:
for f be PartFunc of REAL,REAL, 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,REAL, a be Real;
    assume a in dom f; then
    a in dom(R_EAL f) by MESFUNC5:def 7; then
    consider A be Element of Borel_Sets such that
A1:  A = {a} and
A2:  (R_EAL f) is A-measurable and
A3:  (R_EAL f)|A is_integrable_on B-Meas and
A4:  Integral(B-Meas,(R_EAL f)|A) = 0 by Th45;
    take A;
    thus A = {a} by A1;
    thus f is A-measurable by A2,MESFUNC6:def 1;
    (R_EAL f)|A = f|A by MESFUNC5:def 7; then
    (R_EAL f)|A = (R_EAL(f|A)) by MESFUNC5:def 7;
    hence thesis by A3,A4,MESFUNC6:def 3,def 4;
end;
