
theorem Th25:
for I,J be Subset of REAL, K be non empty closed_interval Subset of REAL,
 x,y be Element of REAL,
 f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 Pg1 be PartFunc of REAL,REAL, E be Element of L-Field
 st x in I & y in J & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:]
  & f = g & Pg1 = ProjPMap1(|.R_EAL g.|,[x,y]) & E = K holds
   Pg1 is E-measurable
proof
    let I,J be Subset of REAL, K be non empty closed_interval Subset of REAL,
    x,y be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Pg1 be PartFunc of REAL,REAL, E be Element of L-Field;
    assume that
A1: x in I & y in J and
A2: dom f = [:[:I,J:],K:] and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: Pg1 = ProjPMap1(|.R_EAL g.|,[x,y]) and
A6: E = K;

    [x,y] in [:I,J:] by A1,ZFMISC_1:87; then
A7:dom Pg1 = K by A2,A4,A5,MESFUN16:27;

    Pg1|K is bounded & Pg1 is_integrable_on K by A1,A2,A3,A4,A5,Th24;
    hence Pg1 is E-measurable by A6,A7,MESFUN14:49;
end;
