
theorem Th51:
for I be Subset of REAL, J be non empty closed_interval Subset of REAL,
 f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 g be PartFunc of [:REAL,REAL:],REAL
 st [:I,J:] = dom f & f is_continuous_on [:I,J:] & f = g holds
   Integral2(L-Meas,|.R_EAL g.|)|I is PartFunc of REAL,REAL
 & Integral2(L-Meas,R_EAL g)|I is PartFunc of REAL,REAL
proof
    let I be Subset of REAL, J be non empty closed_interval Subset of REAL,
    f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL;
    assume that
A1: [:I,J:] = dom f and
A2: f is_continuous_on [:I,J:] and
A3: f = g;

    set Rg =R_EAL g;
    set F = Integral2(L-Meas,|.Rg.|);
    reconsider FI = F|I as PartFunc of REAL,ExtREAL;
A4:dom F = REAL by FUNCT_2:def 1;

    now let y be object;
     assume y in rng FI; then
     consider x be Element of REAL such that
A5: x in dom FI & y = FI.x by PARTFUN1:3;
     reconsider Pg = ProjPMap1(|.Rg.|,x) as PartFunc of REAL,REAL by Th30;
     (Integral2(L-Meas,|.Rg.|)).x = integral(Pg,J)
       by A5,A4,A1,A2,A3,Th46; then
     (Integral2(L-Meas,|.Rg.|)).x in REAL by XREAL_0:def 1;
     hence y in REAL by A5,A4,FUNCT_1:49;
    end; then
    rng FI c= REAL & dom FI c= REAL;
    hence Integral2(L-Meas,|.R_EAL g.|)|I is PartFunc of REAL,REAL
      by RELSET_1:4;

    set G = Integral2(L-Meas,Rg);
    reconsider GI = G|I as PartFunc of REAL,ExtREAL;
A6:dom G = REAL by FUNCT_2:def 1;

    now let y be object;
     assume y in rng GI; then
     consider x be Element of REAL such that
A7: x in dom GI & y= GI.x by PARTFUN1:3;
     reconsider Pg = ProjPMap1(Rg,x) as PartFunc of REAL,REAL by Th30;
     (Integral2(L-Meas,R_EAL g)).x = integral(Pg,J)
       by A7,A6,A1,A2,A3,Th41; then
     (Integral2(L-Meas,Rg)).x in REAL by XREAL_0:def 1;
     hence y in REAL by A7,A6,FUNCT_1:49;
    end; then
    rng GI c= REAL & dom GI c= REAL;
    hence Integral2(L-Meas,R_EAL g)|I is PartFunc of REAL,REAL by RELSET_1:4;
end;
