reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;
reserve pN, qN for Element of NAT^omega;
reserve seq1,seq2,seq3,seq4 for Real_Sequence,
  r,s,e for Real,
  Fr,Fr1, Fr2 for XFinSequence of REAL;

theorem Th50:
  for Fr ex absFr be XFinSequence of REAL st dom absFr=dom Fr &
  |.Sum Fr.| <= Sum absFr & for i st i in dom absFr holds absFr.i=|.Fr.i.|
proof
  let Fr;
  defpred P[object,object] means $2=|.Fr.$1.|;
A1: for i st i in Segm len Fr ex x be Element of REAL st P[i,x]
     proof let i;
      assume i in Segm len Fr;
       consider x being Real such that
A2:     P[i,x];
       reconsider x as Element of REAL by XREAL_0:def 1;
      P[i,x] by A2;
     hence thesis;
    end;
  consider absFr be XFinSequence of REAL such that
A3: dom absFr = Segm len Fr and
A4: for i st i in Segm len Fr holds P[i,absFr.i] from STIRL2_1:sch 5(A1);
  defpred Q[Nat] means $1 <= len Fr implies |.Sum (Fr|$1).| <= Sum
  (absFr|$1);
A5: for i st Q[i] holds Q[i+1]
  proof
    let i such that
A6: Q[i];
    set i1=i+1;
    assume
A7: i1<=len Fr;
    then i < len Fr by NAT_1:13;
    then
A8: i in dom Fr by AFINSQ_1:86;
    then Sum(Fr|i1)=Fr.i+Sum(Fr|i) & absFr.i=|.Fr.i.| by A4,AFINSQ_2:65;
    then
A9: |.Sum(Fr|i1).|<=absFr.i +|.Sum (Fr|i).| by COMPLEX1:56;
    Sum(absFr|i1)=absFr.i+Sum (absFr|i) by A3,A8,AFINSQ_2:65;
    then absFr.i + |.Sum (Fr|i).| <= Sum(absFr|i1) by A6,A7,NAT_1:13,XREAL_1:7;
    hence thesis by A9,XXREAL_0:2;
  end;
  take absFr;
A10: Q[0] by COMPLEX1:44;
  for i holds Q[i] from NAT_1:sch 2(A10,A5);
  then |.Sum (Fr|(len Fr)).| <= Sum (absFr|(len Fr));
  hence thesis by A3,A4;
end;
