
theorem Th40:
for I,J,K be non empty closed_interval Subset of REAL,
  u be Element of [:REAL,REAL:],
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g being PartFunc of [:[:REAL,REAL:],REAL:],REAL
st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
holds Integral2(L-Meas,|.R_EAL g.|).u < +infty
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    u be Element of [:REAL,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;

    dom(Integral2(L-Meas,|.R_EAL g.|)) = [:REAL,REAL:] by FUNCT_2:def 1; then
A4: dom(Integral2(L-Meas,|.R_EAL g.|)| [: I,J:]) = [:I,J:];

    reconsider G = Integral2(L-Meas,|.R_EAL g.|)| [:I,J:]
       as PartFunc of [:REAL,REAL:],REAL by A1,A2,A3,Th32;

    per cases;
    suppose
A5:  u in [:I,J:]; then
     Integral2(L-Meas,|.R_EAL g.|).u = G.u by FUNCT_1:49; then
     Integral2(L-Meas,|.R_EAL g.|).u in rng G by A5,A4,FUNCT_1:3;
     hence Integral2(L-Meas,|.R_EAL g.|).u < +infty by XXREAL_0:9;
    end;
    suppose not u in [:I,J:]; then
     Integral2(L-Meas,|.R_EAL g.|).u = 0 by A1,A3,Lm4;
     hence Integral2(L-Meas,|.R_EAL g.|).u < +infty
       by XREAL_0:def 1,XXREAL_0:9;
    end;
end;
