reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem Th12:
  for seq, seq1 st for n holds seq1.n = seq.0 holds Partial_Sums(
  seq^\1) = (Partial_Sums(seq)^\1) - seq1
proof
  let seq, seq1;
  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 Def1
      .= 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 Def1
    .= seq.(0 + 1) + seq.0 - seq.0 by Def1
    .= seq.(0 + 1) + (seq.0 - seq.0) by RLVECT_1:def 3
    .= seq.(0 + 1) + 09(X) by RLVECT_1:15
    .= seq.(0 + 1)
    .= (seq^\1).0 by NAT_1:def 3;
  hence thesis by A2,Def1;
end;
