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 Th47:
  (seq1 (##) seq2).0=seq1.0*seq2.0
proof
  set S=seq1.0 * seq2.0;
  consider Fr such that
A1: dom Fr = (0 qua Nat)+1 and
A2: for n st n in (0 qua Nat)+1 holds Fr.n = seq1.n * seq2.(0-'n) and
A3: Sum Fr = (seq1 (##) seq2).0 by Def4;
A4: 0-'0=0 & len Fr=1 by A1,XREAL_1:232;
  0 in Segm 1 by NAT_1:44;
  then Fr.0=seq1.0 * seq2.(0-'0) by A2;
  then Fr=<%S%> by A4,AFINSQ_1:34;
  hence thesis by A3,AFINSQ_2:53;
end;
