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;
reserve c for FinSequence of COMPLEX;

theorem
  f is_simple_func_in S iff ex F be Finite_Sep_Sequence of S, c be
  FinSequence of COMPLEX st dom f = union rng F & dom F = dom c & (for n be Nat
st n in dom F for x be set st x in F.n holds (Re f).x = Re c.n) & for n be Nat
  st n in dom F for x be set st x in F.n holds (Im f).x = Im c.n
proof
  hereby
    assume f is_simple_func_in S;
    then consider
    F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX such
    that
A1: F,c are_Re-presentation_of f by Th45;
    F,Im c are_Re-presentation_of Im f by A1,Th48;
    then
A2: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x =
    Im c.n;
    F,Re c are_Re-presentation_of Re f by A1,Th48;
    then
A3: for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x =
    Re c.n;
    dom f = union rng F & dom F = dom c by A1;
    hence ex F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX st dom
f = union rng F & dom F = dom c & (for n be Nat st n in dom F for x be set st x
in F.n holds (Re f).x = Re c.n) & for n be Nat st n in dom F for x be set st x
    in F.n holds (Im f).x = Im c.n by A3,A2;
  end;
  given F be Finite_Sep_Sequence of S, c be FinSequence of COMPLEX such that
A4: dom f = union rng F and
A5: dom F = dom c and
A6: for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x
  = Re c.n and
A7: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x
  = Im c.n;
A8: dom Im f = union rng F by A4,COMSEQ_3:def 4;
  len Im c = len c by COMPLSP2:48;
  then dom Im c = Seg len c by FINSEQ_1:def 3;
  then
A9: dom F = dom Im c by A5,FINSEQ_1:def 3;
  len Re c = len c by COMPLSP2:48;
  then dom Re c = Seg len c by FINSEQ_1:def 3;
  then
A10: dom F = dom Re c by A5,FINSEQ_1:def 3;
A11: dom Re f = union rng F by A4,COMSEQ_3:def 3;
  for n be Nat st n in dom F for x be set st x in F.n holds f.x = c.n
  proof
    let n be Nat;
    assume
A12: n in dom F;
    let x be set;
    assume
A13: x in F.n;
A14: F.n c= union rng F by A12,MESFUNC3:7;
    then x in dom Im f by A8,A13;
    then
A15: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
    x in dom Re f by A11,A13,A14;
    then
A16: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
    (Im f).x = Im c.n by A7,A12,A13;
    then
A17: Im(f.x) = Im(c.n) by A9,A12,A15,Th47;
    (Re f).x = Re c.n by A6,A12,A13;
    then Re(f.x) = Re(c.n) by A10,A12,A16,Th46;
    hence thesis by A17;
  end;
  then F,c are_Re-presentation_of f by A4,A5;
  hence thesis by Th45;
end;
