reserve x, y for set;

theorem Th14:
  for X, N be XFinSequence of INT st len N = len X + 1
  for c be Integer st (N | len X) = c (#) X
  holds Sum N = c * Sum X + N.(len X)
  proof
    let X, N be XFinSequence of INT;
    assume
A1: len N = len X + 1;
    let c be Integer;
    assume
A2: (N | len X) = c (#) X;
A3:    len X in Segm len N by A1,NAT_1:45;
    thus Sum N = Sum (N | (len N))
         .= Sum (N | len X) + N.(len X) by A1,AFINSQ_2:65,A3
         .= c * Sum X + N.(len X) by A2,AFINSQ_2:64;
  end;
