reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

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;
