reserve i,j,k,n for Nat;
reserve x,x1,x2,x3,y1,y2,y3 for set;

theorem Th17:
  the InternalRel of Necklace n = {[i,i+1] where i is Element of
  NAT:i+1 < n} \/ {[i+1,i] where i is Element of NAT:i+1 < n}
proof
  set I = {[i,i+1] where i is Element of NAT:i+1 < n};
  I is Relation-like
  proof
    let x be object;
    assume x in I;
    then ex i being Element of NAT st x = [i,i+1] & i+1<n;
    hence thesis;
  end;
  then reconsider I as Relation;
  set B = n-SuccRelStr;
  deffunc F(Element of NAT) = $1;
  deffunc G(Element of NAT) = $1+1;
  defpred P[Element of NAT] means $1+1 < n;
  set R = {[G(i),F(i)] where i is Element of NAT: P[i]};
A1: I = {[F(i),G(i)] where i is Element of NAT: P[i]};
A2: I~ = R from Convers(A1);
  the InternalRel of B = I by Def6;
  hence thesis by A2,Def7;
end;
