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 Th46:
  (a (#) Rseq) * seq = a * (Rseq * seq)
proof
  let n be Element of NAT;
  thus ((a (#) Rseq) * seq).n = (a (#) Rseq).n * seq.n by Def7
    .= (a * Rseq.n) * seq.n by SEQ_1:9
    .= a * (Rseq.n * seq.n) by RLVECT_1:def 7
    .= a * (Rseq * seq).n by Def7
    .= (a * (Rseq * seq)).n by NORMSP_1:def 5;
end;
