
theorem Th97:
  for X be non empty set, S be SigmaField of X, M be
  sigma_Measure of S, f be PartFunc of X,ExtREAL, A be Element of S st f
  is_integrable_on M holds integral+(M,max+(f|A)) <= integral+(M,max+ f) &
  integral+(M,max-(f|A)) <= integral+(M,max- f) & f|A is_integrable_on M
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, A be Element of S;
A1: max+f is nonnegative by Lm1;
  assume
A2: f is_integrable_on M;
  then consider E be Element of S such that
A3: E = dom f and
A4: f is E-measurable;
A5: max+f is E-measurable by A4,MESFUNC2:25;
A6: f is (E/\A)-measurable by A4,MESFUNC1:30,XBOOLE_1:17;
  dom f /\ (E/\A) = E/\A by A3,XBOOLE_1:17,28;
  then f|(E/\A) is (E/\A)-measurable by A6,Th42;
  then f|E|A is (E/\A)-measurable by RELAT_1:71;
  then
A7: f|A is (E/\A)-measurable by A3,GRFUNC_1:23;
A8: integral+(M,max-f) < +infty by A2;
A9: max-f is nonnegative by Lm1;
A10: integral+(M,max+f) < +infty by A2;
A11: max+f|(E/\A) = (max+f|E)|A by RELAT_1:71;
A12: dom f = dom(max+f) by MESFUNC2:def 2;
  then max+f|E = max+f by A3,GRFUNC_1:23;
  then
A13: integral+(M,max+f|A) <= integral+(M,max+f) by A3,A5,A12,A1,A11,Th83,
XBOOLE_1:17;
  then integral+(M,max+(f|A)) <= integral+(M,max+f) by Th28;
  then
A14: integral+(M,max+(f|A)) < +infty by A10,XXREAL_0:2;
A15: max-f is E-measurable by A3,A4,MESFUNC2:26;
A16: max-f|(E/\A) = (max-f|E)|A by RELAT_1:71;
A17: dom f=dom(max-f) by MESFUNC2:def 3;
  then max-f|E = max-f by A3,GRFUNC_1:23;
  then
A18: integral+(M,max-f|A) <= integral+(M,max-f) by A3,A15,A17,A9,A16,Th83,
XBOOLE_1:17;
  then integral+(M,max-(f|A)) <= integral+(M,max-f) by Th28;
  then
A19: integral+(M,max-(f|A)) < +infty by A8,XXREAL_0:2;
  E /\ A = dom(f|A) by A3,RELAT_1:61;
  hence thesis by A13,A18,A7,A14,A19,Th28;
end;
