reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th20:
  ((f|n)/^1) is not empty implies len ((f|n)/^1) < n
  proof
    assume (f|n)/^1 is not empty; then
    (f|n) is not empty; then
    A2: 1 in dom (f|n) by FINSEQ_5:6;
    (f|n) = ((f|n)|1)^((f|n)/^1) by RFINSEQ:8; then
    len (f|n) = len ((f|n)|1) + len((f|n)/^1) by FINSEQ_1:22; then
    A3: len ((f|n)/^1) + 1 = len (f|n) by A2,Th10;
    len (f|n) <= n by FINSEQ_5:17;
    hence thesis by A3,NAT_1:13;
  end;
