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
  Rseq * (- seq) = (- Rseq) * seq
proof
  let n be Element of NAT;
  thus (Rseq * (- seq)).n = Rseq.n * (-seq).n by Def7
    .= Rseq.n * (-(seq.n)) by BHSP_1:44
    .= (-(Rseq.n)) * seq.n by RLVECT_1:24
    .= (- Rseq).n * seq.n by SEQ_1:10
    .= ((- Rseq) * seq).n by Def7;
end;
