
theorem Th39:
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
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 nonnegative
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 being PartFunc of [:[:REAL,REAL:],REAL:],REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

    dom(R_EAL g) = [:[:I,J:],K:] by A1,A3,MESFUNC5:def 7; then
A4: dom |.R_EAL g.| = [:[:I,J:],K:] by MESFUNC1:def 10;

    now let z be Element of REAL;
     per cases;
     suppose
A5:   z in K;

      reconsider Pg = ProjPMap2(|.R_EAL g.|,z)
       as PartFunc of [:REAL,REAL:],REAL by MESFUN16:30;
      reconsider Pf = Pg as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
      reconsider IJ = [:I,J:] as
        Element of sigma measurable_rectangles(L-Field,L-Field) by MESFUN16:11;
A6:   dom Pf = [:I,J:] by A1,A3,A5,MESFUN16:28;

      Pf is_continuous_on dom Pf by A1,A2,A3,Th20; then
A7:   Pg is_integrable_on Prod_Measure(L-Meas,L-Meas)
    & Integral(Prod_Measure(L-Meas,L-Meas),Pg)
       = Integral(L-Meas,Integral2(L-Meas,R_EAL Pg)) by A6,MESFUN16:57;

      R_EAL Pg = ProjPMap2(|.R_EAL g.|,z) by MESFUNC5:def 7; then
A8:   Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z
       = Integral(Prod_Measure(L-Meas,L-Meas),R_EAL Pg) by MESFUN12:def 7;

A9:   Pg is IJ-measurable by A1,A2,A3,A5,Th27;

      for u being object st u in dom(Pg|IJ) holds 0 <= (Pg|IJ).u
      proof
       let u be object;
       assume
A10:   u in dom(Pg|IJ); then
       u in IJ; then
       reconsider u as Element of [:REAL,REAL:];

A11:   (ProjPMap2(|.R_EAL g.|,z)).u = (|.R_EAL g.|).(u,z)
         by A5,A10,A4,ZFMISC_1:87,MESFUN12:def 4;
A12:   (R_EAL g).([u,z]) = g.([u,z]) by MESFUNC5:def 7;

       (|.R_EAL g.|).(u,z) = |.(R_EAL g).([u,z]).|
         by A4,A5,A10,ZFMISC_1:87,MESFUNC1:def 10; then
       (|.R_EAL g.|).(u,z) = |. g.([u,z]) .| by A12,EXTREAL1:12;
       hence thesis by A10,A11,FUNCT_1:49;
      end;
      hence 0 <= Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z
        by A6,A7,A8,A9,MESFUNC6:52,84;
     end;
     suppose not z in K;
      hence 0 <= Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z
        by A1,A3,Lm5;
     end;
    end;
    hence thesis;
end;
