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 + seq2).n - seq3.n by NORMSP_1:def 3
    .= (seq1.n + seq2.n) - seq3.n by NORMSP_1:def 2
    .= seq1.n + (seq2.n - seq3.n) by RLVECT_1:def 3
    .= seq1.n + (seq2 - seq3).n by NORMSP_1:def 3
    .= (seq1 + (seq2 - seq3)).n by NORMSP_1:def 2;
end;
