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;
