reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th15:
(X = dom f & for x st x in dom f holds 0 = f.x) implies
  f is_integrable_on M & Integral(M,f) =0
proof
   assume A1:X=dom f & for x st x in dom f holds 0 = f.x;
   X is Element of S by MEASURE1:7; then
   R_EAL f is_integrable_on M & Integral(M,R_EAL f) = 0 by A1,LPSPACE1:22;
   hence thesis;
end;
