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
  a * (seq1 - seq2) = a * seq1 - a * seq2
proof
  now
    let n be Element of NAT;
    thus (a * (seq1 - seq2)).n = a * (seq1 - seq2).n by CLVECT_1:def 14
      .= a * (seq1.n - seq2.n) by NORMSP_1:def 3
      .= a * seq1.n - a * seq2.n by CLVECT_1:9
      .= (a * seq1).n - a * seq2.n by CLVECT_1:def 14
      .= (a * seq1).n - (a * seq2).n by CLVECT_1:def 14
      .= (a * seq1 - a * seq2).n by NORMSP_1:def 3;
  end;
  hence thesis by FUNCT_2:63;
end;
