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 Th12:
  (for n holds seq1.n = seq.0) implies Partial_Sums(seq^\1) = (
  Partial_Sums(seq)^\1) - seq1
proof
  assume
A1: for n holds seq1.n = seq.0;
A2: now
    let n;
    thus ((Partial_Sums(seq)^\1) - seq1).(n + 1) = (Partial_Sums(seq)^\1).(n +
    1) - seq1.(n + 1) by NORMSP_1:def 3
      .= (Partial_Sums(seq)^\1).(n + 1) - seq.0 by A1
      .= Partial_Sums(seq).(n + 1 + 1) - seq.0 by NAT_1:def 3
      .= seq.(n+1+1) + Partial_Sums(seq).(n + 1) - seq.0 by BHSP_4:def 1
      .= seq.(n+1+1) + Partial_Sums(seq).(n + 1) - seq1.n by A1
      .= seq.(n+1+1) + (Partial_Sums(seq).(n + 1) - seq1.n) by RLVECT_1:def 3
      .= seq.(n+1+1) + ((Partial_Sums(seq)^\1).n - seq1.n) by NAT_1:def 3
      .= seq.(n+1+1) + ((Partial_Sums(seq)^\1) - seq1).n by NORMSP_1:def 3
      .= ((Partial_Sums(seq)^\1) - seq1).n + (seq^\1).(n + 1) by NAT_1:def 3;
  end;
  ((Partial_Sums(seq)^\1) - seq1).0 = (Partial_Sums(seq)^\1).0 - seq1.0 by
NORMSP_1:def 3
    .= (Partial_Sums(seq)^\1).0 - seq.0 by A1
    .= Partial_Sums(seq).(0 + 1) - seq.0 by NAT_1:def 3
    .= Partial_Sums(seq).0 + seq.(0 + 1) - seq.0 by BHSP_4:def 1
    .= seq.(0 + 1) + seq.0 - seq.0 by BHSP_4:def 1
    .= seq.(0 + 1) + (seq.0 - seq.0) by RLVECT_1:def 3
    .= seq.(0 + 1) + 09(X) by RLVECT_1:15
    .= seq.(0 + 1) by RLVECT_1:4
    .= (seq^\1).0 by NAT_1:def 3;
  hence thesis by A2,BHSP_4:def 1;
end;
