reserve p, q, x, y for Real,
  n for Nat;

theorem Th8:
  for f being real-valued FinSequence holds (x-y) * f = x*f - y*f
proof
  let f be real-valued FinSequence;
A1: dom ((x-y)*f) = dom f by VALUED_1:def 5;
A2: dom (x*f-y*f) = dom(x*f) /\ dom(y*f) by VALUED_1:12;
A3: dom (x*f) = dom f by VALUED_1:def 5;
A4: dom (y*f) = dom f by VALUED_1:def 5;
  now
    let n;
    assume
A5: n in dom ((x-y)*f);
    thus ((x-y) * f).n = (x-y)*(f.n) by VALUED_1:6
    .= x*(f.n)-y*(f.n)
    .= x*f.n-(y*f).n by VALUED_1:6
    .= (x*f).n-(y*f).n by VALUED_1:6
    .= (x*f - y*f).n by A1,A2,A3,A4,A5,VALUED_1:13;
  end;
  hence thesis by A1,A2,A3,A4;
end;
