reserve x, y for set;

theorem
  for F be XFinSequence of INT for c be Integer holds
  c * Sum F = Sum ((c (#) F) | (len F -' 1)) + c * F.(len F -' 1)
  proof
    let F be XFinSequence of INT;
    let c be Integer;
    per cases;
    suppose len F = 0; then
A1:    F is empty & F.(len F -' 1) = 0 by FUNCT_1:def 2;
      then Sum F = 0;
      hence thesis by A1;
    end;
    suppose len F > 0; then
A2:    len F -' 1 + 1 = len F by NAT_1:14,XREAL_1:235;
A3:    dom F = dom (c (#) F) by VALUED_1:def 5;
A4:    c * Sum F = Sum (c (#) F) by AFINSQ_2:64;
A5:    Sum (c (#) F) = Sum((c (#) F) | len F) by A3;
      len F -' 1 in Segm len F by A2,NAT_1:45;
      then Sum ((c (#) F) | (len F -' 1 + 1)) =
        (Sum ((c (#) F) | (len F -' 1))) + (c (#) F).(len F -' 1)
                                              by A3,AFINSQ_2:65;
      hence thesis by A4,A5,A2,VALUED_1:6;
    end;
  end;
