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

theorem Th1:
  for X being Abelian add-associative 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 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 2
      .= 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:def 3
      .= PSseq1.n + (seq1.(n + 1) + seq2.(n + 1)) + PSseq2.n by RLVECT_1:def 3
      .= PSseq1.n + (seq1 + seq2).(n + 1) + PSseq2.n by NORMSP_1:def 2
      .= PSseq1.n + PSseq2.n + (seq1 + seq2).(n + 1) by RLVECT_1:def 3
      .= (PSseq1 + PSseq2).n + (seq1 + seq2).(n + 1) by NORMSP_1:def 2;
  end;
  (PSseq1 + PSseq2).0 = PSseq1.0 + PSseq2.0 by NORMSP_1:def 2
    .= seq1.0 + PSseq2.0 by Def1
    .= seq1.0 + seq2.0 by Def1
    .= (seq1 + seq2).0 by NORMSP_1:def 2;
  hence thesis by A1,Def1;
end;
