
theorem Th41:
for I be Subset of REAL, J be non empty closed_interval Subset of REAL,
    x be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, Pg1 be PartFunc of REAL,REAL
 st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g &
    Pg1 = ProjPMap1(R_EAL g,x) holds Pg1 is_integrable_on L-Meas
  & integral(Pg1,J) = Integral(L-Meas,Pg1)
  & integral(Pg1,J) = Integral(L-Meas,(ProjPMap1(R_EAL g,x)))
  & integral(Pg1,J) = (Integral2(L-Meas,R_EAL g)).x
proof
    let I be Subset of REAL, J be non empty closed_interval Subset of REAL,
    x be Element of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL,
    Pg1 be PartFunc of REAL,REAL;
    assume that
A1:  x in I and
A2:  dom f = [:I,J:] and
A3:  f is_continuous_on [:I,J:] and
A4:  f = g and
A5:  Pg1 = ProjPMap1(R_EAL g,x);

A6:J is Element of L-Field by MEASUR10:5,MEASUR12:75;
A7:dom Pg1 = J by A1,A2,A4,A5,Th27;
    Pg1|J is bounded & Pg1 is_integrable_on J by A1,A2,A3,A4,A5,Th40;
    hence Pg1 is_integrable_on L-Meas
  & integral(Pg1,J) = Integral(L-Meas,Pg1) by A6,A7,MESFUN14:49;
    hence integral(Pg1,J) = Integral(L-Meas,ProjPMap1(R_EAL g,x))
      by A5,MESFUNC5:def 7;
    hence thesis by MESFUN12:def 8;
end;
