reserve s for set,
  i,j for Nat,
  c,c1,c2,c3 for Complex,
  F,F1,F2 for complex-valued FinSequence,
  R,R1,R2 for i-element complex-valued FinSequence;

theorem
  (c1 + c2)*F = c1*F + c2*F
proof
A1: dom ((c1 + c2)*F) = dom F by VALUED_1:def 5;
A2: dom (c1*F + c2*F) = dom (c1*F) /\ dom (c2*F) by VALUED_1:def 1;
A3: dom (c1*F) = dom F by VALUED_1:def 5;
A4: dom (c2*F) = dom F by VALUED_1:def 5;
    now
      let i;
      assume A5: i in dom ((c1+c2)*F);
      thus ((c1+c2)*F).i = (c1+c2)*(F.i) by VALUED_1:6
      .= c1*(F.i) + c2*(F.i)
      .= c1*(F.i) + (c2*F).i by VALUED_1:6
      .= (c1*F).i + (c2*F).i by VALUED_1:6
      .= (c1*F+c2*F).i by A1,A2,A3,A4,A5,VALUED_1:def 1;
    end;
    hence thesis by A1,A2,A3,A4,FINSEQ_1:13;
end;
