
theorem Th40:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,REAL, A,B,E be Element of S
  st E = dom f & f is E-measurable & f is nonpositive & A c= B
  holds Integral(M,f|A) >= Integral(M,f|B)
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,REAL, A,B,E be Element of S;
    assume that
A1:  E = dom f and
A2:  f is E-measurable and
A3:  f is nonpositive and
A4:  A c= B;

A5:  E = dom(R_EAL f) by A1,MESFUNC5:def 7;
A6:  R_EAL f is E-measurable by A2,MESFUNC6:def 1;

     for x be set st x in dom (R_EAL f) holds (R_EAL f).x <= 0
     proof
      let x be set;
      assume x in dom(R_EAL f);
      f.x <= 0 by A3,MESFUNC6:53;
      hence (R_EAL f).x <= 0 by MESFUNC5:def 7;
     end; then
     Integral(M,(R_EAL f)|A) >= Integral(M,(R_EAL f)|B)
       by A5,A6,A4,MESFUNC5:9,MESFUN11:64; then
     Integral(M,R_EAL (f|A)) >= Integral(M,(R_EAL f)|B) by Th16; then
     Integral(M,R_EAL (f|A)) >= Integral(M,R_EAL(f|B)) by Th16; then
     Integral(M,f|A) >= Integral(M,R_EAL(f|B)) by MESFUNC6:def 3;
     hence Integral(M,f|A) >= Integral(M,f|B) by MESFUNC6:def 3;
end;
