
theorem Th42:
for I be non empty closed_interval Subset of REAL, J be Subset of REAL,
  y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
  g be PartFunc of [:REAL,REAL:],REAL, Pg2 be PartFunc of REAL,REAL
 st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g &
    Pg2 = ProjPMap2(R_EAL g,y) holds Pg2|I is bounded & Pg2 is_integrable_on I
proof
    let I be non empty closed_interval Subset of REAL, J be Subset of REAL,
    y be Element of REAL, f be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    g be PartFunc of [:REAL,REAL:],REAL, Pg2 be PartFunc of REAL,REAL;
    assume that
A1:  y in J and
A2:  dom f = [:I,J:] and
A3:  f is_continuous_on [:I,J:] and
A4:  f = g and
A5:  Pg2 = ProjPMap2(R_EAL g,y);

    Pg2 = R_EAL(ProjPMap2(g,y)) by A5,Th31; then
A6: Pg2 = ProjPMap2(g,y) by MESFUNC5:def 7;

    dom Pg2 = I by A1,A2,A4,A5,Th28;
    hence thesis by A6,A2,A3,A4,Th33,INTEGRA5:10,11;
end;
