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 Th93:
  Rotate(Rotate(f,p),p) = Rotate(f,p)
proof
  per cases;
  suppose
A1: p in rng f;
    then reconsider f9 = f as non empty FinSequence of D;
A2: (Rotate(f,p))/.1 = p by A1,Th92;
    then
A3: Rotate(f9,p):-p = Rotate(f,p) by Th44;
A4: p in rng Rotate(f,p) by A1,Th91;
A5: len<*p*> = 1 by FINSEQ_1:39;
    Rotate(f9,p)-:p = <*p*> by A2,Th44;
    hence Rotate(Rotate(f,p),p) = Rotate(f,p)^(<*p*>/^1) by A3,A4,Def2
      .= Rotate(f,p)^{} by A5,FINSEQ_5:32
      .= Rotate(f,p) by FINSEQ_1:34;
  end;
  suppose
    not p in rng f;
    hence thesis by Def2;
  end;
end;
