reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for r be Real, n be non zero Nat
   ex f be FinSequence of REAL st len f = n & Sum f = r
proof
  let r be Real, n be non zero Nat;
  reconsider k = n-1 as Nat;
  0 in REAL by XREAL_0:def 1; then
  reconsider g = k|-> 0 as FinSequence of REAL by FINSEQ_2:63;
  reconsider h = g^<*r*> as FinSequence of REAL by RVSUM_1:145;
  A1: len (g^<*r*>)  = (len g)+1 by FINSEQ_2:16
  .= k + 1;
  Sum (g^<*r*>) = Sum(k|-> 0) + r by RVSUM_1:74
  .= 0 + r; then
  Sum h = r & len h = k+1 by A1;
  hence thesis;
end;
