
theorem Th35:
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
  Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|) is Function of REAL,REAL
& Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K is PartFunc of REAL,REAL
& Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) is Function of REAL,REAL
& Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g)|K is PartFunc of 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 = Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|);
    set RF = Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g);

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

    now let q be object;
     assume q in rng F; then
     consider z be Element of REAL such that
A6:  z in dom F & q = F.z by PARTFUN1:3;
     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;

     per cases;
     suppose A7: z in K; then
A8:   dom Pf = [:I,J:] by A1,A3,MESFUN16:28; then
      Pf is_continuous_on [:I,J:] by A1,A2,A3,Th20; then
      reconsider G2 = Integral2(L-Meas,R_EAL Pg)|I
        as PartFunc of REAL,REAL by A8,MESFUN16:51;
      Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z
       = Integral(Prod_Measure(L-Meas,L-Meas),Pg) by A1,A2,A3,A7,Th28
      .= integral(G2,I) by A1,A2,A3,A8,Th20,MESFUN16:58;
      hence q in REAL by A6,XREAL_0:def 1;
     end;
     suppose not z in K; then
      dom ProjPMap2(|.R_EAL g.|,z) = {} by A1,A3,MESFUN16:28; then
      Integral(Prod_Measure(L-Meas,L-Meas),ProjPMap2(|.R_EAL g.|,z)) = 0
        by MESFUN16:1; then
      Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|).z = 0
        by MESFUN12:def 7;
      hence q in REAL by A6,XREAL_0:def 1;
     end;
    end; then
    rng F c= REAL;
    hence Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)
     is Function of REAL,REAL by A4,RELSET_1:4;
    hence Integral1(Prod_Measure(L-Meas,L-Meas),|.R_EAL g.|)|K
     is PartFunc of REAL,REAL by PARTFUN1:11;

    now let q be object;
     assume q in rng RF; then
     consider z be Element of REAL such that
A9:  z in dom RF & q = RF.z by PARTFUN1:3;
     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;

     per cases;
     suppose A10: z in K; then
A11:  dom Pf = [:I,J:] by A1,A3,MESFUN16:28; then
      Pf is_continuous_on [:I,J:] by A1,A2,A3,Th18; then
      reconsider G2 = Integral2(L-Meas,R_EAL Pg)|I
        as PartFunc of REAL,REAL by A11,MESFUN16:51;
      Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).z
       = Integral(Prod_Measure(L-Meas,L-Meas),Pg) by A1,A2,A3,A10,Th23
      .= integral(G2,I) by A1,A2,A3,A11,Th18,MESFUN16:58;
      hence q in REAL by A9,XREAL_0:def 1;
     end;
     suppose not z in K; then
      dom ProjPMap2(R_EAL g,z) = {} by A1,A3,MESFUN16:28; then
      Integral(Prod_Measure(L-Meas,L-Meas),ProjPMap2(R_EAL g,z)) = 0
        by MESFUN16:1; then
      Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g).z = 0 by MESFUN12:def 7;
      hence q in REAL by A9,XREAL_0:def 1;
     end;
    end; then
    rng RF c= REAL;
    hence Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g)
     is Function of REAL,REAL by A5,RELSET_1:4;
    hence Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g)|K
     is PartFunc of REAL,REAL by PARTFUN1:11;
end;
