reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;

theorem
  m ==> n iff m = 0
  proof
    the reduction of ARS_02 = [:{0},{0,1,2}:] by Def19; then
    m ==> n iff m in {0} & n in {0,1,2} by ZFMISC_1:87; then
    m ==> n iff m = 0 & (n = 0 or n = 1 or n = 2)
    by TARSKI:def 1,ENUMSET1:def 1;
    hence thesis by ThB1;
  end;
