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
  seq is summable implies z * seq is summable & Sum(z*seq) = z * Sum(seq )
proof
  assume seq is summable;
  then
A1: Partial_Sums(seq) is convergent;
  then z * Partial_Sums(seq) is convergent by CLVECT_2:5;
  then Partial_Sums(z * seq) is convergent by Th3;
  hence z * seq is summable;
  thus Sum(z * seq) = lim (z * Partial_Sums(seq)) by Th3
    .= z * Sum(seq) by A1,CLVECT_2:15;
end;
