
theorem Th27:
for I,J be non empty closed_interval Subset of REAL, K be Subset of REAL,
  z be Element of REAL,
  f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
  g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
  Pg2 be PartFunc of [:REAL,REAL:],REAL,
  E be Element of sigma measurable_rectangles(L-Field,L-Field)
  st z in K & dom f = [:[:I,J:],K:] & f is_continuous_on [:[:I,J:],K:]
   & f = g & Pg2 = ProjPMap2(|.R_EAL g.|,z) & E = [:I,J:] holds
   Pg2 is E-measurable
proof
    let I,J be non empty closed_interval Subset of REAL, K be Subset of REAL,
    z be Element of REAL,
    f be PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    Pg2 be PartFunc of [:REAL,REAL:],REAL,
    E be Element of sigma measurable_rectangles(L-Field,L-Field);
    assume that
A1: z in K and
A2: dom f = [:[:I,J:],K:] and
A3: f is_continuous_on [:[:I,J:],K:] and
A4: f = g and
A5: Pg2 = ProjPMap2(|.R_EAL g.|,z) and
A6: E = [:I,J:];

    reconsider Pf2 = Pg2 as PartFunc of [:RNS_Real,RNS_Real:],RNS_Real;
    dom Pf2 = [:I,J:] by A1,A2,A4,A5,MESFUN16:28;
    hence Pg2 is E-measurable by A2,A3,A4,A5,A6,Th20,MESFUN16:50;
end;
