
theorem Th37:
  for X be non empty set, S be SigmaField of X, M be sigma_Measure
  of S, f be PartFunc of X,ExtREAL st f is_simple_func_in S holds dom f is
  Element of S
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let f be PartFunc of X,ExtREAL;
  assume f is_simple_func_in S;
  then ex F be Finite_Sep_Sequence of S st dom f = union rng F & for n being
Nat,x,y being Element of X st n in dom F & x in F.n & y in F.n holds f.x = f.y
  by MESFUNC2:def 4;
  hence thesis by MESFUNC2:31;
end;
