theorem
  (ex A be Element of S st A = dom f & f is A-measurable ) implies (f
  is_integrable_on M iff abs f is_integrable_on M)
proof
  assume ex A be Element of S st A = dom f & f is A-measurable;
  then consider A be Element of S such that
A1: A = dom f and
A2: f is A-measurable;
  R_EAL f is A-measurable by A2;
  then R_EAL f is_integrable_on M iff |. R_EAL f .| is_integrable_on M by A1,
MESFUNC5:100;
  then f is_integrable_on M iff R_EAL abs f is_integrable_on M by Th1;
  hence thesis;
end;
