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
  seq1 + (seq2 + seq3) = (seq1 + seq2) + seq3
proof
  let n be Element of NAT;
  thus (seq1 + (seq2 + seq3)).n = seq1.n + (seq2 + seq3).n by NORMSP_1:def 2
    .= seq1.n + (seq2.n + seq3.n) by NORMSP_1:def 2
    .= (seq1.n + seq2.n) + seq3.n by RLVECT_1:def 3
    .= (seq1 + seq2).n + seq3.n by NORMSP_1:def 2
    .= ((seq1 + seq2) + seq3).n by NORMSP_1:def 2;
end;
