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;
