
theorem Th40:
for I be Subset of REAL, J be non empty closed_interval Subset of REAL,
  x be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
  g be PartFunc of [:REAL,REAL:],REAL, Pg1 be PartFunc of REAL,REAL
 st x in I & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g &
    Pg1 = ProjPMap1(R_EAL g,x) holds Pg1|J is bounded & Pg1 is_integrable_on J
proof
    let I be Subset of REAL, J be non empty closed_interval Subset of REAL,
    x be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, Pg1 be PartFunc of REAL,REAL;
    assume that
A1: x in I and
A2: dom f = [:I,J:] and
A3: f is_continuous_on [:I,J:] and
A4: f = g and
A5: Pg1 = ProjPMap1(R_EAL g,x);

    Pg1 = R_EAL(ProjPMap1(g,x)) by A5,Th31; then
A6: Pg1 = ProjPMap1(g,x) by MESFUNC5:def 7;

    dom Pg1 = J by A1,A2,A4,A5,Th27;
    hence thesis by A6,A2,A3,A4,Th33,INTEGRA5:10,11;
end;
