
theorem Th96:
  for X be non empty set, S be SigmaField of X, M be
sigma_Measure of S, f be PartFunc of X,ExtREAL st f is_integrable_on M holds 0
  <= integral+(M,max+f) & 0 <= integral+(M,max-f) & -infty < Integral(M,f) &
  Integral(M,f) < +infty
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL such that
A1: f is_integrable_on M;
  consider A be Element of S such that
A2: A = dom f and
A3: f is A-measurable by A1;
A4: integral+(M,max+f) <> +infty by A1;
A5: dom f=dom(max+f) by MESFUNC2:def 2;
A6: max+f is nonnegative by Lm1;
  then -infty <> integral+(M,max+f) by A2,A3,A5,Th79,MESFUNC2:25;
  then reconsider maxf1=integral+(M,max+f) as Element of REAL
           by A4,XXREAL_0:14;
A7: max+f is A-measurable by A3,MESFUNC2:25;
A8: integral+(M,max-f) <> +infty by A1;
A9: dom f=dom(max-f) by MESFUNC2:def 3;
A10: max-f is nonnegative by Lm1;
  then -infty <> integral+(M,max-f) by A2,A3,A9,Th79,MESFUNC2:26;
  then reconsider maxf2=integral+(M,max-f) as Element of REAL
           by A8,XXREAL_0:14;
  integral+(M,max+f)-integral+(M,max-f) = maxf1-maxf2 by SUPINF_2:3;
  hence thesis by A2,A3,A5,A9,A6,A10,A7,Th79,MESFUNC2:26,XXREAL_0:9,12;
end;
