
theorem Th26:
for I,J be Subset of REAL, K be non empty closed_interval Subset of REAL,
 x,y be Element of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL, Pg1 be PartFunc of REAL,REAL
 st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:]
  & f = g & Pg1 = ProjPMap1(|.R_EAL g.|,[x,y])
  holds
    Pg1 is_integrable_on L-Meas
   & integral(Pg1,K) = Integral(L-Meas,Pg1)
   & integral(Pg1,K) = Integral(L-Meas,ProjPMap1(|.R_EAL g.|,[x,y]))
   & integral(Pg1,K) = (Integral2(L-Meas,|.R_EAL g.|)).([x,y])
proof
    let I,J be Subset of REAL, K be non empty closed_interval Subset of REAL,
    x,y be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Pg1 be PartFunc of REAL,REAL;
    assume that
A1:  x in I & y in J and
A2:  dom f = [:[:I,J:],K:] and
A3:  f is_continuous_on [:[:I,J:],K:] and
A4:  f = g and
A5:  Pg1 = ProjPMap1(|.R_EAL g.|,[x,y]);

A6: K is Element of L-Field by MEASUR10:5,MEASUR12:75;

    [x,y] in [:I,J:] by A1,ZFMISC_1:87; then
A7:dom Pg1 = K by A2,A4,A5,MESFUN16:27;

    Pg1|K is bounded & Pg1 is_integrable_on K by A1,A2,A3,A4,A5,Th24;
    hence
A8: Pg1 is_integrable_on L-Meas
  & integral(Pg1,K) = Integral(L-Meas,Pg1) by A6,A7,MESFUN14:49;

    R_EAL(Pg1) = ProjPMap1(|.R_EAL g.|,[x,y]) by A5,MESFUNC5:def 7;
    hence thesis by A8,MESFUN12:def 8;
end;
