
theorem Th21:
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
 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]) holds
   Pg1|K is bounded & Pg1 is_integrable_on K
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;
    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]);

    [x,y] in [:I,J:] by A1,ZFMISC_1:87; then
    dom Pg1 = K by A2,A4,A5,MESFUN16:27;
    hence Pg1|K is bounded & Pg1 is_integrable_on K
      by A2,A3,A4,A5,Th17,INTEGRA5:10,11;
end;
