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 * (seq1 + seq2) = Rseq * seq1 + Rseq * seq2
proof
  let n be Element of NAT;
  thus (Rseq * (seq1 + seq2)).n = Rseq.n * (seq1 + seq2).n by Def7
    .= Rseq.n * (seq1.n + seq2.n) by NORMSP_1:def 2
    .= Rseq.n * seq1.n + Rseq.n * seq2.n by RLVECT_1:def 5
    .= (Rseq * seq1).n + Rseq.n * seq2.n by Def7
    .= (Rseq * seq1).n + (Rseq * seq2).n by Def7
    .= (Rseq * seq1 + Rseq * seq2).n by NORMSP_1:def 2;
end;
