
theorem Th33:
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
 f is_simple_func_in COM(S,M)
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL;
    assume A1: f is_simple_func_in S; then
A2: f is real-valued by MESFUNC2:def 4;
    consider F be Finite_Sep_Sequence of S such that
A3:  dom f = union rng F and
A4:  for n be Nat, x,y be Element of X st
      n in dom F & x in F.n & y in F.n holds f.x = f.y by A1,MESFUNC2:def 4;
    reconsider F1=F as Finite_Sep_Sequence of COM(S,M) by Th32;
    dom f = union rng F1 by A3;
    hence f is_simple_func_in COM(S,M) by A2,A4,MESFUNC2:def 4;
end;
