reserve X for set;
reserve X,X1,X2 for non empty set;
reserve S for SigmaField of X;
reserve S1 for SigmaField of X1;
reserve S2 for SigmaField of X2;
reserve M for sigma_Measure of S;
reserve M1 for sigma_Measure of S1;
reserve M2 for sigma_Measure of S2;

theorem
for E being Element of S, f being E-measurable PartFunc of X,ExtREAL
  st dom f c= E & f is_a.e.integrable_on M holds f is_integrable_on M
proof
   let E be Element of S, f be E-measurable PartFunc of X,ExtREAL;
   assume that
A1: dom f c= E and
A2: f is_a.e.integrable_on M;
   reconsider E1 = dom f as Element of S by A2,Th16;
   consider A be Element of S such that
A3: M.A = 0 & A c= dom f & f|A` is_integrable_on M by A2;
A4:f|A` = f|(dom f \ A) by Th15; then
A5:dom(f|A`) = dom f /\ (dom f \ A) by RELAT_1:61; then
A6:dom(f|A`) = (E1 /\ E1) \ A by XBOOLE_1:49 .= E1 \ A; then
A7:dom(max+(f|A`)) = E1\A & dom(max-(f|A`)) = E1\A by MESFUNC2:def 2,def 3;
A8:f is E1-measurable by A1,MESFUNC1:30; then
   f is (E1\A)-measurable by XBOOLE_1:36,MESFUNC1:30; then
   f|A` is (E1\A)-measurable by A4,A5,A6,MESFUNC5:42; then
A9:max+(f|A`) is (E1\A)-measurable & max-(f|A`) is (E1\A)-measurable
     by A6,MESFUNC2:25,26;
A10:E1 = dom(max+f) & E1 = dom(max-f) by MESFUNC2:def 2,def 3;
A11:max+f is nonnegative & max-f is nonnegative by MESFUN11:5;
   Integral(M,max+f|(dom f \ A)) = Integral(M,max+f)
 & Integral(M,max-f|(dom f \ A)) = Integral(M,max-f)
     by A3,A8,A10,MESFUNC2:25,26,MESFUNC5:95; then
   Integral(M,max+(f|A`)) = Integral(M,max+f)
 & Integral(M,max-(f|A`)) = Integral(M,max-f) by A4,MESFUNC5:28; then
   integral+(M,max+(f|A`)) = Integral(M,max+f)
 & integral+(M,max-(f|A`)) = Integral(M,max-f)
      by A7,A9,MESFUNC5:88,MESFUN11:5; then
A12:integral+(M,max+(f|A`)) = integral+(M,max+f)
 & integral+(M,max-(f|A`)) = integral+(M,max-f)
     by A8,A10,A11,MESFUNC2:25,26,MESFUNC5:88;
   integral+(M,max+(f|A`)) < +infty & integral+(M,max-(f|A`)) < +infty
     by A3,MESFUNC5:def 17;
   hence f is_integrable_on M by A8,A12,MESFUNC5:def 17;
end;
