reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;
reserve x,A for set;

theorem Th45:
  f is_simple_func_in S iff ex F be Finite_Sep_Sequence of S, a be
  FinSequence of COMPLEX st F,a are_Re-presentation_of f
proof
  hereby
    assume f is_simple_func_in S;
    then consider
    F being Finite_Sep_Sequence of S, a be FinSequence of COMPLEX
    such that
A1: dom f = union rng F & dom F= dom a & for n be Nat st n in dom F
    for x be set st x in F.n holds f.x=a.n by Th44;
    take F,a;
    thus F,a are_Re-presentation_of f by A1;
  end;
  given F being Finite_Sep_Sequence of S, a be FinSequence of COMPLEX such
  that
A2: F,a are_Re-presentation_of f;
A3: 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
  proof
    let n being Nat,x,y being Element of X;
    assume that
A4: n in dom F and
A5: x in F.n and
A6: y in F.n;
    f.x=a.n by A2,A4,A5;
    hence thesis by A2,A4,A6;
  end;
  dom f = union rng F by A2;
  hence thesis by A3;
end;
