theorem
  z1 * Partial_Sums(seq1) + z2 * Partial_Sums(seq2) = Partial_Sums(z1*
  seq1 + z2*seq2)
proof
  thus z1 * Partial_Sums(seq1) + z2 * Partial_Sums(seq2) = Partial_Sums(z1 *
  seq1) + z2 * Partial_Sums(seq2) by Th3
    .= Partial_Sums(z1 * seq1) + Partial_Sums(z2 * seq2) by Th3
    .= Partial_Sums(z1 * seq1 + z2 * seq2) by Th1;
end;
