
theorem Th48:
for f be PartFunc of REAL,REAL st f is_integrable_on B-Meas
  holds f is_integrable_on L-Meas & Integral(B-Meas,f) = Integral(L-Meas,f)
proof
    let f be PartFunc of REAL,REAL;
    assume f is_integrable_on B-Meas; then
A1: R_EAL f is_integrable_on B-Meas by MESFUNC6:def 4;
    hence f is_integrable_on L-Meas
      by Th38,MEASUR12:def 11,def 12,MESFUNC6:def 4;
    Integral(B-Meas,f) = Integral(B-Meas,R_EAL f)
  & Integral(L-Meas,f) = Integral(L-Meas,R_EAL f) by MESFUNC6:def 3;
    hence Integral(B-Meas,f) = Integral(L-Meas,f)
      by A1,Th38,MEASUR12:def 11,def 12;
end;
