
theorem Th48:
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,
  E be Element of L-Field
 st y in J & dom f = [:I,J:] & f is_continuous_on [:I,J:] & f = g
  & Pg2 = ProjPMap2(|.R_EAL g.|,y) & E = I holds Pg2 is E-measurable
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,
    E be Element of L-Field;
    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) and
A6:  E = I;

A7: dom Pg2 = I by A1,A2,A4,A5,Th28;
    Pg2|I is bounded & Pg2 is_integrable_on I by A1,A2,A3,A4,A5,Th47;
    hence thesis by A7,A6,MESFUN14:49;
end;
