
theorem Th43:
for I be non empty closed_interval Subset of REAL, J be Subset of REAL,
  y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
  g be PartFunc of [:REAL,REAL:],REAL, Pg2 be PartFunc of REAL,REAL
 st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g &
    Pg2 = ProjPMap2(R_EAL g,y) holds Pg2 is_integrable_on L-Meas
  & integral(Pg2,I) = Integral(L-Meas,Pg2)
  & integral(Pg2,I) = Integral(L-Meas,(ProjPMap2(R_EAL g,y)))
  & integral(Pg2,I) = (Integral1(L-Meas,R_EAL g)).y
proof
    let I be non empty closed_interval Subset of REAL, J be Subset of REAL,
    y be Element of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL,
    Pg2 be PartFunc of REAL,REAL;
    assume that
A1:  y in J and
A2:  dom f = [:I,J:] and
A3:  f is_continuous_on [:I,J:] and
A4:  f = g and
A5:  Pg2 = ProjPMap2(R_EAL g,y);

A6:I is Element of L-Field by MEASUR10:5,MEASUR12:75;
A7:dom Pg2 = I by A1,A2,A4,A5,Th28;
    Pg2|I is bounded & Pg2 is_integrable_on I by A1,A2,A3,A4,A5,Th42;
    hence Pg2 is_integrable_on L-Meas
  & integral(Pg2,I) = Integral(L-Meas,Pg2) by A6,A7,MESFUN14:49;
    hence integral(Pg2,I) = Integral(L-Meas,ProjPMap2(R_EAL g,y))
      by A5,MESFUNC5:def 7;
    hence thesis by MESFUN12:def 7;
end;
