reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve  n for Nat;

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