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
  empty_rel(D) = {}
proof
  assume
A1: not thesis;
  set x = the Element of empty_rel(D);
  empty_rel(D) is Subset of D* by Def7;
  then x in D* by A1,TARSKI:def 3;
  then reconsider a = x as FinSequence of D by FINSEQ_1:def 11;
  a in empty_rel(D) by A1;
  hence contradiction by Def9;
end;
