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

theorem Th24:
  for i,j being Nat st (i = j + 1 or j = i + 1) & i in the carrier
  of Necklace n & j in the carrier of Necklace n holds [i,j] in the InternalRel
  of Necklace n
proof
  let i,j be Nat such that
A1: i = j + 1 or j = i + 1 and
A2: i in the carrier of Necklace n and
A3: j in the carrier of Necklace n;
  per cases by A1;
  suppose
A4: i=j+1;
    then [j,j+1] in the InternalRel of Necklace n by A2,Th20,Th21;
    then [j+1,j] in (the InternalRel of Necklace n)~ by RELAT_1:def 7;
    hence thesis by A4,RELAT_2:13;
  end;
  suppose
    j=i+1;
    hence thesis by A3,Th20,Th21;
  end;
end;
