
theorem Th60:
for y be Element of REAL, 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,
  P2Gz be PartFunc of REAL,REAL st
  y in J & [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
& P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I
 holds P2Gz||I is bounded & P2Gz is_integrable_on I
proof
    let y be Element of REAL;
    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,
    P2Gz be PartFunc of REAL,REAL;
    assume that
A1: y in J and
A2: [:[:I,J:],K:] = dom f and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: P2Gz = ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I;

    reconsider F2 = Integral2(L-Meas,R_EAL g)| [:I,J:]
      as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real by A2,A3,A4,Th32;
    reconsider G2 = F2 as PartFunc of [:REAL,REAL:],REAL;

A6: dom(Integral2(L-Meas,R_EAL g)) = [:REAL,REAL:] by FUNCT_2:def 1;

    Y-section([:I,J:],y) = I by A1,MEASUR11:22; then
A7: ProjPMap2(Integral2(L-Meas,R_EAL g)| [:I,J:],y)
     = ProjPMap2(Integral2(L-Meas,R_EAL g),y)|I by MESFUN12:34;

A8: Integral2(L-Meas,R_EAL g)| [:I,J:] = R_EAL (G2| [:I,J:]) by MESFUNC5:def 7;

    F2 is_uniformly_continuous_on [:I,J:] by A2,A3,A4,Th34; then
    F2 is_continuous_on [:I,J:] by NFCONT_2:7;
    hence P2Gz||I is bounded & P2Gz is_integrable_on I
      by A1,A5,A6,A7,A8,MESFUN16:42;
end;
