reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;

theorem Th2:
  for X being Abelian add-associative right_zeroed
  right_complementable non empty addLoopStr, seq1, seq2 being sequence of X
  holds Partial_Sums(seq1) - Partial_Sums(seq2) = Partial_Sums(seq1 - seq2)
proof
  let X be Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr, seq1, seq2 be sequence of X;
  set PSseq1 = Partial_Sums(seq1);
  set PSseq2 = Partial_Sums(seq2);
A1: now
    let n;
    thus (PSseq1 - PSseq2).(n + 1) = PSseq1.(n + 1) - PSseq2.(n + 1) by
NORMSP_1:def 3
      .= (PSseq1.n + seq1.(n + 1)) - PSseq2.(n + 1) by Def1
      .= (PSseq1.n + seq1.(n + 1)) - (seq2.(n + 1) + PSseq2.n) by Def1
      .= ((PSseq1.n + seq1.(n + 1)) - seq2.(n + 1)) - PSseq2.n by RLVECT_1:27
      .= (PSseq1.n + (seq1.(n + 1) - seq2.(n + 1))) - PSseq2.n by
RLVECT_1:def 3
      .= (PSseq1.n - PSseq2.n) + (seq1.(n + 1) - seq2.(n + 1)) by
RLVECT_1:def 3
      .= (PSseq1 - PSseq2).n + (seq1.(n + 1) - seq2.(n + 1)) by NORMSP_1:def 3
      .= (PSseq1 - PSseq2).n + (seq1 - seq2).(n + 1) by NORMSP_1:def 3;
  end;
  (PSseq1 - PSseq2).0 = (PSseq1).0 - (PSseq2).0 by NORMSP_1:def 3
    .= seq1.0 - (PSseq2).0 by Def1
    .= seq1.0 - seq2.0 by Def1
    .= (seq1 - seq2).0 by NORMSP_1:def 3;
  hence thesis by A1,Def1;
end;
