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;

theorem
  for f being circular non trivial FinSequence of D holds rng(f/^1) = rng f
proof
  let f be circular non trivial FinSequence of D;
  thus rng(f/^1) c= rng f by FINSEQ_5:33;
  let x be object;
  f = <*f/.1*>^(f/^1) by FINSEQ_5:29;
  then
A1: rng f = rng<*f/.1*> \/ rng(f/^1) by FINSEQ_1:31;
  assume
A2: x in rng f;
  per cases by A1,A2,XBOOLE_0:def 3;
  suppose
    x in rng<*f/.1*>;
    then x in {f/.1} by FINSEQ_1:39;
    then x = f/.1 by TARSKI:def 1;
    then
A3: x = f/.len f by Def1A;
A4: len f >= 1+1 by NAT_D:60;
    then len f >= 1 by XXREAL_0:2;
    then
A5: len(f/^1) = len f - 1 by RFINSEQ:def 1;
    then 1 <= len(f/^1) by A4,XREAL_1:19;
    then
A6: len(f/^1) in dom(f/^1) by FINSEQ_3:25;
    len(f/^1) + 1 = len f by A5;
    then x = (f/^1)/.len(f/^1) by A3,A6,FINSEQ_5:27;
    hence thesis by A6,PARTFUN2:2;
  end;
  suppose
    x in rng(f/^1);
    hence thesis;
  end;
end;
