reserve x,z for set;
reserve k for Element of NAT;
reserve D for non empty set;
reserve X for set;
reserve p,r for relation;
reserve a,a1,a2,b for FinSequence;
reserve a,b for FinSequence of D;
reserve p,r for Element of relations_on D;

theorem
  for x,y being Element of D holds {<*x,y*>} is Element of relations_on D
proof
  let x,y be Element of D;
A1: for a1,a2 being FinSequence of D st a1 in {<*x,y*>} & a2 in {<*x,y*>}
  holds len a1 = len a2
  proof
    let a1,a2 be FinSequence of D;
    assume that
A2: a1 in {<*x,y*>} and
A3: a2 in {<*x,y*>};
    a1 = <*x,y*> by A2,TARSKI:def 1;
    hence thesis by A3,TARSKI:def 1;
  end;
  <*x*>^<*y*> is FinSequence of D;
  then <*x,y*> in D* by FINSEQ_1:def 11;
  then {<*x,y*>} c= D* by ZFMISC_1:31;
  hence thesis by A1,Def7;
end;
