
theorem Th58:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be nonpositive PartFunc of X,ExtREAL st f is_simple_func_in S
  holds Integral(M,f) = -integral'(M,-f) & Integral(M,f) = -integral'(M,max-f)
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   f be nonpositive PartFunc of X,ExtREAL;
   assume A1: f is_simple_func_in S; then
   reconsider A = dom f as Element of S by MESFUNC5:37;
A2:f is A-measurable by A1,MESFUNC2:34;
   integral+(M,-f) = integral'(M,-f) by A1,Th30,MESFUNC5:77;
   hence A3: Integral(M,f) = - integral'(M,-f) by A2,Th57;
   f = -(max-f) by Th32;
   hence Integral(M,f) = - integral'(M,max-f) by A3,Th36;
end;
