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
  (Rseq1 + Rseq2) * seq = Rseq1 * seq + Rseq2 * seq
proof
  let n be Element of NAT;
  thus ((Rseq1 + Rseq2) * seq).n = (Rseq1 + Rseq2).n * seq.n by Def7
    .= (Rseq1.n + Rseq2.n) * seq.n by SEQ_1:7
    .= Rseq1.n * seq.n + Rseq2.n * seq.n by RLVECT_1:def 6
    .= (Rseq1 * seq).n + Rseq2.n * seq.n by Def7
    .= (Rseq1 * seq).n + (Rseq2 * seq).n by Def7
    .= (Rseq1 * seq + Rseq2 * seq).n by NORMSP_1:def 2;
end;
