
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,
  Gxy be PartFunc of REAL,REAL st
  [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
& Gxy = (Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g))|K
 holds Gxy||K is bounded & Gxy is_integrable_on K
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,
    Gxy be PartFunc of REAL,REAL;
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: Gxy = (Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g))|K;

    dom Integral1(Prod_Measure(L-Meas,L-Meas),R_EAL g) = REAL
      by FUNCT_2:def 1; then
    dom Gxy = K by A4;
    hence thesis by A1,A2,A3,A4,Th37,INTEGRA5:10,11;
end;
