reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;
reserve a, a1, a2 for TwoValued Alternating FinSequence;
reserve fs, fs1, fs2 for FinSequence of X,
  fss, fss2 for Subset of fs;
reserve F, F1 for FinSequence of INT,
  k, m, n, ma for Nat;
reserve i,j,k,m,n for Nat,
  D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve D for non empty set,
  p for Element of D,
  f for FinSequence of D;
reserve f for circular FinSequence of D;
reserve i, i1, i2, j, k for Nat;
reserve D for non empty set,
  f1 for FinSequence of D;

theorem Th17:
  for f1 being FinSequence st len f1<k holds mid(f1,k,k)={}
proof
  let f1 be FinSequence;
  assume
A1: len f1<k;
  then len f1+1<=k by NAT_1:13;
  then
A2: len f1+1-1<=k-1 by XREAL_1:9;
  0+1<=k by A1,NAT_1:13;
  then len f1<=k-'1 by A2,XREAL_1:233;
  then
A3: f1/^(k-'1)={} by FINSEQ_5:32;
  k-'k+1=k-k+1 by XREAL_1:233
    .=1;
  then mid(f1,k,k)=(f1/^(k-'1))|1 by Def3;
  hence thesis by A3;
end;
