
theorem Th32:
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 be PartFunc of [:[:REAL,REAL:],REAL:],REAL
 st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
 holds
  Integral2(L-Meas,|.R_EAL g.|) is Function of [:REAL,REAL:],REAL
& Integral2(L-Meas,|.R_EAL g.|)| [:I,J:] is PartFunc of [:REAL,REAL:],REAL
& Integral2(L-Meas,R_EAL g) is Function of [:REAL,REAL:],REAL
& Integral2(L-Meas,R_EAL g)| [:I,J:] is PartFunc of [:REAL,REAL:],REAL
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;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

    set F = Integral2(L-Meas,|.R_EAL g.|);
    set RF = Integral2(L-Meas,R_EAL g);

A4: dom F = [:REAL,REAL:] by FUNCT_2:def 1;
A5: dom RF = [:REAL,REAL:] by FUNCT_2:def 1;

    now let q be object;
     assume q in rng F; then
     consider p be Element of [:REAL,REAL:] such that
A6:  p in dom F & q=F.p by PARTFUN1:3;
     consider x,y be object such that
A7:  x in REAL & y in REAL & p = [x,y] by ZFMISC_1:84;
     reconsider x,y as Element of REAL by A7;
     reconsider Pg = ProjPMap1(|.R_EAL g.|,[x,y]) as PartFunc of REAL,REAL
       by MESFUN16:30;

     per cases;
     suppose p in [:I,J:]; then
      x in I & y in J by A7,ZFMISC_1:87; then
      Integral2(L-Meas,|.R_EAL g.|).([x,y]) = integral(Pg,K) by A1,A2,A3,Th26;
      hence q in REAL by A6,A7,XREAL_0:def 1;
     end;
     suppose not p in [:I,J:]; then
      Integral2(L-Meas,|.R_EAL g.|).p = 0 by A1,A3,Lm4;
      hence q in REAL by A6,XREAL_0:def 1;
      end;
    end; then
    rng F c= REAL;
    hence Integral2(L-Meas,|.R_EAL g.|)
      is Function of [:REAL,REAL:],REAL by A4,RELSET_1:4;
    hence Integral2(L-Meas,|.R_EAL g.|)| [:I,J:]
      is PartFunc of [:REAL,REAL:],REAL by PARTFUN1:11;

    now let q be object;
     assume q in rng RF; then
     consider p be Element of [:REAL,REAL:] such that
A8:  p in dom RF & q=RF.p by PARTFUN1:3;
     consider x,y be object such that
A9: x in REAL & y in REAL & p = [x,y] by ZFMISC_1:84;
     reconsider x,y as Element of REAL by A9;
     reconsider Pg = ProjPMap1(R_EAL g,[x,y]) as PartFunc of REAL,REAL
       by MESFUN16:30;

     per cases;
     suppose p in [:I,J:]; then
      x in I & y in J by A9,ZFMISC_1:87; then
      Integral2(L-Meas,R_EAL g).([x,y]) = integral(Pg,K) by A1,A2,A3,Th22;
      hence q in REAL by A8,A9,XREAL_0:def 1;
     end;
     suppose not p in [:I,J:]; then
      Integral2(L-Meas,R_EAL g).p = 0 by A1,A3,Lm4;
      hence q in REAL by A8,XREAL_0:def 1;
     end;
    end; then
    rng RF c= REAL;
    hence Integral2(L-Meas,R_EAL g) is Function of [:REAL,REAL:],REAL
      by A5,RELSET_1:4;
    hence thesis by PARTFUN1:11;
end;
