
theorem
for y be Element of REAL, I,J,K be non empty closed_interval Subset of REAL,
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
  P2Gz be PartFunc of REAL,REAL st
  y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
& P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I holds
   ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I is_integrable_on L-Meas
 & integral(P2Gz,I) = Integral(L-Meas,ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I)
 & integral(P2Gz,I) = Integral(L-Meas,ProjPMap2(Integral2(L-Meas,R_EAL g),y))
 & integral(P2Gz,I) = (Integral1(L-Meas,Integral2(L-Meas,R_EAL g))).y
proof
    let y be Element of REAL;
    let I,J,K be non empty closed_interval Subset of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    P2Gz be PartFunc of REAL,REAL;
    assume that
A1: y in J and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I;

    set Gz = Integral2(L-Meas,R_EAL g);

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

A7: ProjPMap2(Gz,y) is_integrable_on L-Meas by A2,A3,A4,Th48;
    hence ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I is_integrable_on L-Meas
      by A6,MESFUNC5:97;

A8: dom(Integral2(L-Meas,R_EAL g)) = [:REAL,REAL:] by FUNCT_2:def 1;
    [#]REAL = REAL by SUBSET_1:def 3; then
    dom(ProjPMap2(Integral2(L-Meas,R_EAL g),y)) = REAL by A8,MESFUN16:26; then
A9: dom P2Gz = I by A5;
    P2Gz||I is bounded & P2Gz is_integrable_on I by A1,A2,A3,A4,A5,Th60; then
    P2Gz|I is_integrable_on L-Meas
  & Integral(L-Meas,P2Gz|I) = integral(P2Gz||I) by A6,A9,MESFUN14:49;
    hence
A10: integral(P2Gz,I)
      = Integral(L-Meas,ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I)
      by A5,MESFUNC5:def 7;

    REAL in L-Field by PROB_1:5; then
    reconsider NI = REAL \ I as Element of L-Field by A6,PROB_1:6;
    set PGz20 = ProjPMap2(Gz,y)|I;
    set PGz21 = ProjPMap2(Gz,y)|NI;

A11:Integral(L-Meas,PGz21) = 0 by A2,A4,Th55;

    I \/ NI = REAL by XBOOLE_1:45; then
    ProjPMap2(Gz,y)|(I \/ NI) = ProjPMap2(Gz,y); then
    Integral(L-Meas,ProjPMap2(Gz,y))
      = Integral(L-Meas,PGz20) + Integral(L-Meas,PGz21)
       by A6,A7,XBOOLE_1:85,MESFUNC5:98;
    hence integral(P2Gz,I) = Integral(L-Meas,ProjPMap2(Gz,y))
      by A10,A11,XXREAL_3:4;
    hence integral(P2Gz,I) = (Integral1(L-Meas,Integral2(L-Meas,R_EAL g))).y
      by MESFUN12:def 7;
end;
