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;

theorem
  i ==> j iff i = 0
  proof
    the reduction of ARS_01 = [:{0},{0,1}:] by Def18; then
    i ==> j iff i in {0} & j in {0,1} by ZFMISC_1:87; then
    i ==> j iff i = 0 & (j = 0 or j = 1) by TARSKI:def 1,def 2;
    hence thesis by ThA1;
  end;
