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 + b) * seq = a * seq + b * seq
proof
  now
    let n be Element of NAT;
    thus ((a + b) * seq).n = (a + b) * seq.n by CLVECT_1:def 14
      .= a * seq.n + b * seq.n by CLVECT_1:def 3
      .= (a * seq).n + b * seq.n by CLVECT_1:def 14
      .= (a * seq).n + (b * seq).n by CLVECT_1:def 14
      .= (a * seq + b * seq).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
