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 Th3:
  Partial_Sums(z * seq) = z * Partial_Sums(seq)
proof
  set PSseq = Partial_Sums(seq);
A1: now
    let n;
    thus (z * PSseq).(n + 1) = z * PSseq.(n + 1) by CLVECT_1:def 14
      .= z * (PSseq.n + seq.(n + 1)) by BHSP_4:def 1
      .= z * PSseq.n + z * seq.(n + 1) by CLVECT_1:def 2
      .= (z * PSseq).n + z * seq.(n + 1) by CLVECT_1:def 14
      .= (z * PSseq).n + (z * seq).(n + 1) by CLVECT_1:def 14;
  end;
  (z * PSseq).0 = z * PSseq.0 by CLVECT_1:def 14
    .= z * seq.0 by BHSP_4:def 1
    .= (z * seq).0 by CLVECT_1:def 14;
  hence thesis by A1,BHSP_4:def 1;
end;
