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 Th44:
  for Fr1,Fr2 st dom Fr1=dom Fr2 & for n st n in len Fr1 holds Fr1
  .n=r*Fr2.n holds Sum Fr1 = r*Sum Fr2
proof
  let Fr1,Fr2 such that
A1: dom Fr1=dom Fr2 and
A2: for n st n in len Fr1 holds Fr1.n=r*Fr2.n;
A3: Fr1|(dom Fr1)=Fr1 & Fr2|(dom Fr1)=Fr2 by A1;
  defpred P[Nat] means $1 <= len Fr1 implies Sum(Fr1|$1)=r*Sum(Fr2|
  $1);
A4: for i st P[i] holds P[i+1]
  proof
    let i such that
A5: P[i];
    assume
A6: i+1 <= len Fr1;
    then i< len Fr1 by NAT_1:13;
    then
A7: i in len Fr1 by AFINSQ_1:86;
    then
A8: Fr1.i=r*Fr2.i by A2;
    Sum(Fr1|(i+1))=Fr1.i+Sum(Fr1|i) & Sum(Fr2|(i+1))=Fr2.i+Sum(Fr2|i) by A1,A7,
AFINSQ_2:65;
    hence thesis by A5,A6,A8,NAT_1:13;
  end;
  A9: P[0];
  for i holds P[i] from NAT_1:sch 2(A9,A4);
  hence thesis by A3;
end;
