
theorem Th29:
for I,J,K be non empty closed_interval Subset of REAL,
 f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
 g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
 E be Element of sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
 st [:[:I,J:],K:] = dom f & f is_continuous_on [:[:I,J:],K:] & f = g
  & E = [:[:I,J:],K:] holds g is E-measurable
proof
    let I,J,K be non empty closed_interval Subset of REAL,
    f being PartFunc of [:[:RNS_Real,RNS_Real:],RNS_Real:],RNS_Real,
    g be PartFunc of [:[:REAL,REAL:],REAL:],REAL,
    E be Element of sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field);
    assume that
A1: [:[:I,J:],K:] = dom f and
A2: f is_continuous_on [:[:I,J:],K:] and
A3: f = g and
A4: E = [:[:I,J:],K:];

    for r being Real holds E /\ (less_dom (g,r))
     in sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
    proof
     let r be Real;
A5:  less_dom(g,r) = g"(].-infty,r.[) by MESFUN16:17;
     consider H be Subset of [:RNS_Real,RNS_Real,RNS_Real:] such that
A6:   H /\ E = f"(].-infty,r.[) & H is open by A1,A2,A4,MESFUN16:24;
     H in sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
        by A6,Th7; then
     f"(].-infty,r.[) in sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
      by A6,FINSUB_1:def 2;
     hence E /\ (less_dom (g,r)) in
      sigma measurable_rectangles
      (sigma measurable_rectangles(L-Field,L-Field),L-Field)
        by A3,A5,FINSUB_1:def 2;
    end;
    hence g is E-measurable by MESFUNC6:12;
end;
