
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 be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
  y be Element of REAL
st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
  holds
   ProjPMap2(Integral2(L-Meas,R_EAL g),y) is Function of REAL,REAL
 & ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y) is Function 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,
    y be Element of REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g;

    reconsider F2 = Integral2(L-Meas,R_EAL g)
     as Function of [:REAL,REAL:],REAL by A1,A2,A3,Th32;

A4: dom Integral2(L-Meas,R_EAL g) = [:REAL,REAL:] by FUNCT_2:def 1; then
A5: dom |.Integral2(L-Meas,R_EAL g).| = [:REAL,REAL:] by MESFUNC1:def 10;
    dom F2 = [:REAL,REAL:] by FUNCT_2:def 1; then
A6: dom |.F2.| = [:REAL,REAL:] by VALUED_1:def 11;
A7: [#]REAL = REAL by SUBSET_1:def 3; then
A8: dom ProjPMap2(Integral2(L-Meas,R_EAL g),y) = REAL by A4,MESFUN16:26;
A9: dom ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y) = REAL by A7,A5,MESFUN16:26;
A10:dom ProjPMap2(F2,y) = REAL by A7,A4,MESFUN16:26;
A11:dom ProjPMap2(|.F2.|,y) = REAL by A6,A7,MESFUN16:26;

    now let r be object;
     assume r in rng ProjPMap2(Integral2(L-Meas,R_EAL g),y); then
     consider x be Element of REAL such that
A12: x in dom ProjPMap2(Integral2(L-Meas,R_EAL g),y)
   & r = (ProjPMap2(Integral2(L-Meas,R_EAL g),y)).x by PARTFUN1:3;
A13: [x,y] in [:REAL,REAL:]; then
     [x,y] in dom Integral2(L-Meas,R_EAL g) by FUNCT_2:def 1; then
A14: r = F2.(x,y) by A12,MESFUN12:def 4;
     [x,y] in dom F2 by A13,FUNCT_2:def 1; then
     r = ProjPMap2(F2,y).x by A14,MESFUN12:def 4; then
     r in rng ProjPMap2(F2,y) by A10,FUNCT_1:3;
     hence r in REAL;
    end; then
    rng ProjPMap2(Integral2(L-Meas,R_EAL g),y) c= REAL;
    hence ProjPMap2(Integral2(L-Meas,R_EAL g),y) is Function of REAL,REAL
      by A8,FUNCT_2:2;

    now let r be object;
     assume r in rng ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y); then
     consider x be Element of REAL such that
A15: x in dom ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y)
   & r = (ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y)).x by PARTFUN1:3;
     [x,y] in [:REAL,REAL:]; then
A16: [x,y] in dom Integral2(L-Meas,R_EAL g) by FUNCT_2:def 1; then
A17: [x,y] in dom |.Integral2(L-Meas,R_EAL g).| by MESFUNC1:def 10; then
     r = (|.Integral2(L-Meas,R_EAL g).|).(x,y) by A15,MESFUN12:def 4; then
A18: r = |. Integral2(L-Meas,R_EAL g).(x,y) .| by A17,MESFUNC1:def 10;

     set p = [x,y];
     reconsider p as Element of [:REAL,REAL:];
A19: p in dom |.F2.| by A16,VALUED_1:def 11;
     r = |. F2.p .| by A18,EXTREAL1:12; then
     r = (|.F2.|).(x,y) by A19,VALUED_1:def 11; then
     r = ProjPMap2(|.F2.|,y).x by A19,MESFUN12:def 4; then
     r in rng ProjPMap2(|.F2.|,y) by A11,FUNCT_1:3;
     hence r in REAL;
    end; then
    rng ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y) c= REAL;
    hence ProjPMap2(|.Integral2(L-Meas,R_EAL g).|,y) is Function of REAL,REAL
      by A9,FUNCT_2:2;
end;
