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;

theorem
  for D being non empty set, p being Element of D,
      f being FinSequence of D st p in rng f holds
        len(f:-p) <= len f
proof
  let D be non empty set, p be Element of D, f be FinSequence of D;
  assume
A1: p in rng f;
  then 1 <= p..f by FINSEQ_4:21;
  then
A2: len f - 1 >= len f - p..f by XREAL_1:10;
  len (f:-p) = len f - p..f + 1 by A1,FINSEQ_5:50;
  then len (f:-p) - 1 = len f - p..f;
  hence thesis by A2,XREAL_1:9;
end;
