reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
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
  now
    let n be Element of NAT;
    thus (seq1 - (seq2 - seq3)).n = seq1.n - (seq2 - seq3).n by NORMSP_1:def 3
      .= seq1.n - (seq2.n - seq3.n) by NORMSP_1:def 3
      .= (seq1.n - seq2.n) + seq3.n by RLVECT_1:29
      .= (seq1 - seq2).n + seq3.n by NORMSP_1:def 3
      .= ((seq1 - seq2) + seq3).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
