
theorem Th44:
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);

    dom Pg1 = J by A1,A2,A4,A5,Th27;
    hence thesis by A2,A3,A4,A5,Th38,INTEGRA5:10,11;
end;
