
theorem
for I,J,K be non empty closed_interval Subset of REAL,
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g being PartFunc of [:[:REAL,REAL:],REAL:],REAL,
  E be Element of L-Field
st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
 holds Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|) is E-measurable
proof
    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,
    E be Element of L-Field;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

    set F = Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|);
    set F0 = F|K;

A4: dom F = REAL by FUNCT_2:def 1; then
A5: dom F0 = K;

    reconsider G = F0 as PartFunc of REAL,REAL by A1,A2,A3,Th35;
    reconsider GG = G as PartFunc of RNS_Real,RNS_Real;

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

    G|K is bounded & G is_integrable_on K
      by A1,A2,A3,A5,Th36,INTEGRA5:10,11; then
    G is_integrable_on L-Meas by A5,A6,MESFUN14:49; then
A7: F0 is_integrable_on L-Meas by MESFUNC5:def 7;

    reconsider R = REAL as Element of L-Field by PROB_1:5;
    set NK = R \ K;

A8: NK is Element of L-Field by A6,PROB_1:6;

A9: K \/ NK = REAL by XBOOLE_1:45;
A10: K /\ NK = {} by XBOOLE_1:85,XBOOLE_0:def 7;

A11: F is nonnegative by A1,A2,A3,Th39;

    for r being Real holds R /\ (less_dom(F,r)) in L-Field
    proof
     let r be Real;
A12: ex E be Element of L-Field
      st E = dom F0 & F0 is E-measurable by A7,MESFUNC5:def 17;
     per cases;
     suppose
A13:  r <= 0;
      less_dom(F,r) = {}
      proof
       assume less_dom(F,r) <> {}; then
       consider x be object such that
A14:   x in less_dom(F,r) by XBOOLE_0:def 1;
       x in dom F & F.x < r by A14,MESFUNC1:def 11;
       hence contradiction by A11,A13;
      end;
      hence R /\ less_dom(F,r) in L-Field by PROB_1:4;
     end;
     suppose
A15:  0 < r;
      for z be object holds z in less_dom(F,r) iff z in less_dom(F0,r) \/ NK
      proof
       let z be object;
       hereby assume
A16:    z in less_dom(F,r); then
A17:    F.z < r by MESFUNC1:def 11;

A18:    z in NK implies z in less_dom(F0,r) \/ NK by XBOOLE_0:def 3;
        now assume
A19:     z in K; then
         F0.z < r by A17,FUNCT_1:49; then
         z in less_dom(F0,r) by A5,A19,MESFUNC1:def 11;
         hence z in less_dom(F0,r) \/ NK by XBOOLE_0:def 3;
        end;
        hence z in less_dom(F0,r) \/ NK by A9,A16,A18,XBOOLE_0:def 3;
       end;
       assume z in less_dom (F0,r) \/ NK; then
       per cases by XBOOLE_0:def 3;
       suppose
A20:    z in less_dom(F0,r); then
        z in dom F0 & F0.z < r by MESFUNC1:def 11; then
        F.z < r by FUNCT_1:49;
        hence z in less_dom(F,r) by A4,A20,MESFUNC1:def 11;
       end;
       suppose
A21:    z in NK; then
        not z in K by XBOOLE_0:def 5; then
        F.z < r by A15,A1,A3,A21,Lm5;
        hence z in less_dom(F,r) by A4,A21,MESFUNC1:def 11;
       end;
      end; then
      less_dom(F,r) = less_dom(F0,r) \/ NK by TARSKI:2; then
A22:  R /\ less_dom(F,r)
       = K /\ (less_dom(F0,r) \/ NK) \/ NK /\ (less_dom(F0,r) \/ NK)
          by A9,XBOOLE_1:23;

      K /\ (less_dom(F0,r) \/ NK)
       = K /\ less_dom(F0,r) \/ (K /\ NK) by XBOOLE_1:23; then
A23:  K /\ (less_dom(F0,r) \/ NK)
       in L-Field by A4,A10,A12;

A24:  less_dom(F0,r) c= K by A5,MESFUNC1:def 11;

      NK /\ (less_dom(F0,r) \/ NK)
       = (NK /\ less_dom(F0,r)) \/ (NK /\ NK) by XBOOLE_1:23; then
      NK /\ (less_dom(F0,r) \/ NK)
       = {} \/ (NK /\ NK) by A24;
      hence R /\ less_dom(F,r) in L-Field by A8,A22,A23,PROB_1:3;
     end;
    end; then
    F is R-measurable;
    hence F is E-measurable by MESFUNC1:30;
end;
