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 + b) * seq = a * seq + b * seq
proof
  let n be Element of NAT;
  thus ((a + b) * seq).n = (a + b) * seq.n by NORMSP_1:def 5
    .= a * seq.n + b * seq.n by RLVECT_1:def 6
    .= (a * seq).n + b * seq.n by NORMSP_1:def 5
    .= (a * seq).n + (b * seq).n by NORMSP_1:def 5
    .= (a * seq + b * seq).n by NORMSP_1:def 2;
end;
