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

theorem Th3:
  for X being vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty RLSStruct, seq being
  sequence of X holds Partial_Sums(a * seq) = a * Partial_Sums(seq)
proof
  let X be vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty RLSStruct, seq be sequence of X;
  set PSseq = Partial_Sums(seq);
A1: now
    let n;
    thus (a * PSseq).(n + 1) = a * PSseq.(n + 1) by NORMSP_1:def 5
      .= a * (PSseq.n + seq.(n + 1)) by Def1
      .= a * PSseq.n + a * seq.(n + 1) by RLVECT_1:def 5
      .= (a * PSseq).n + a * seq.(n + 1) by NORMSP_1:def 5
      .= (a * PSseq).n + (a * seq).(n + 1) by NORMSP_1:def 5;
  end;
  (a * PSseq).0 = a * PSseq.0 by NORMSP_1:def 5
    .= a * seq.0 by Def1
    .= (a * seq).0 by NORMSP_1:def 5;
  hence thesis by A1,Def1;
end;
