
theorem Th23:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X,ExtREAL, E be Element of S, n be Nat
  st E = dom f & f is nonnegative & f is E-measurable & Integral(M,f) = 0
  holds M.(E /\ great_eq_dom(f,1/(n+1))) = 0
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL, E be Element of S, n be Nat;
    assume that
A1:  E = dom f and
A2:  f is nonnegative and
A3:  f is E-measurable and
A4:  Integral(M,f) = 0;

    assume A5: M.(E /\ great_eq_dom(f,1/(n+1))) <> 0;

    E /\ great_eq_dom(f,1/(n+1)) in S by A1,A3,MESFUNC1:27; then
A6: M.(E /\ great_eq_dom(f,1/(n+1))) > 0 by A5,MEASURE1:def 2;
    great_eq_dom(f,1/(n+1)) c= E by A1,MESFUNC1:def 14; then
A7: M.(great_eq_dom(f,1/(n+1))) > 0 by A6,XBOOLE_1:28;
    1/(n+1) > 0 by XREAL_1:139; then
    1/(n+1)*M.(great_eq_dom(f,1/(n+1))) > 0 by A7;
    hence contradiction by A3,A1,A2,A4,MESFUN13:16;
end;
