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 Th48:
  for F be Finite_Sep_Sequence of S, a be FinSequence of COMPLEX
holds F,a are_Re-presentation_of f iff F,Re a are_Re-presentation_of Re f & F,
  Im a are_Re-presentation_of Im f
proof
  let F be Finite_Sep_Sequence of S, a be FinSequence of COMPLEX;
  hereby
    assume
A1: F,a are_Re-presentation_of f;
    len Im a = len a by COMPLSP2:48;
    then dom Im a = Seg len a by FINSEQ_1:def 3;
    then dom Im a = dom a by FINSEQ_1:def 3;
    then
A2: dom F = dom Im a by A1;
    dom Im f = dom f by COMSEQ_3:def 4;
    then
A3: dom Im f = union rng F by A1;
A4: for n be Nat st n in dom F for x be set st x in F.n holds (Im f).x = Im a.n
    proof
      let n be Nat;
      assume
A5:   n in dom F;
      let x be set;
      assume
A6:   x in F.n;
      F.n c= union rng F by A5,MESFUNC3:7;
      then x in dom Im f by A3,A6;
      then
A7:   (Im f).x = Im(f.x) by COMSEQ_3:def 4;
      Im(f.x) = Im(a.n) by A1,A5,A6;
      hence thesis by A2,A5,A7,Th47;
    end;
    len Re a = len a by COMPLSP2:48;
    then dom Re a = Seg len a by FINSEQ_1:def 3;
    then dom Re a = dom a by FINSEQ_1:def 3;
    then
A8: dom F = dom Re a by A1;
    dom Re f = dom f by COMSEQ_3:def 3;
    then
A9: dom Re f = union rng F by A1;
    for n be Nat st n in dom F for x be set st x in F.n holds (Re f).x = Re a.n
    proof
      let n be Nat;
      assume
A10:  n in dom F;
      let x be set;
      assume
A11:  x in F.n;
      F.n c= union rng F by A10,MESFUNC3:7;
      then x in dom Re f by A9,A11;
      then
A12:  (Re f).x = Re(f.x) by COMSEQ_3:def 3;
      Re(f.x) = Re(a.n) by A1,A10,A11;
      hence thesis by A8,A10,A12,Th46;
    end;
    hence
    F,Re a are_Re-presentation_of Re f & F,Im a are_Re-presentation_of Im
    f by A9,A3,A8,A2,A4;
  end;
  assume that
A13: F,Re a are_Re-presentation_of Re f and
A14: F,Im a are_Re-presentation_of Im f;
A15: dom F = dom Re a by A13;
A16: dom Re f = union rng F by A13;
  then
A17: dom f = union rng F by COMSEQ_3:def 3;
A18: dom F = dom Im a by A14;
A19: dom Im f = union rng F by A14;
A20: for n be Nat st n in dom F for x be set st x in F.n holds f.x = a.n
  proof
    let n be Nat;
    assume
A21: n in dom F;
    let x be set;
    assume
A22: x in F.n;
A23: F.n c= union rng F by A21,MESFUNC3:7;
    then x in dom Im f by A19,A22;
    then
A24: (Im f).x = Im(f.x) by COMSEQ_3:def 4;
    x in dom Re f by A16,A22,A23;
    then
A25: (Re f).x = Re(f.x) by COMSEQ_3:def 3;
    (Im f).x = Im a.n by A14,A21,A22;
    then
A26: Im(f.x) = Im(a.n) by A18,A21,A24,Th47;
    (Re f).x = Re a.n by A13,A21,A22;
    then Re(f.x) = Re(a.n) by A15,A21,A25,Th46;
    hence thesis by A26;
  end;
  len Re a = len a by COMPLSP2:48;
  then dom Re a = Seg len a by FINSEQ_1:def 3;
  then dom F = dom a by A15,FINSEQ_1:def 3;
  hence thesis by A17,A20;
end;
