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;
reserve A for set;

theorem
  x <<01>> y iff x,y are_divergent<=1_wrt the reduction of X
  proof set R = the reduction of X;
    thus x <<01>> y implies x,y are_divergent<=1_wrt R
    proof
      given z such that
A1:   x <=01= z & z =01=> y;
      take z;
      (z ==> x or z = x) & (z ==> y or z = y) by A1;
      hence ([z,x] in R or z = x) & ([z,y] in R or z = y);
    end;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    given a being object such that
A2: ([a,x] in R or a = x) & ([a,y] in R or a = y);
    a in field R or a = x or a = y by A2,RELAT_1:15; then
    reconsider z = a as Element of X by F0;
    take z;
    thus z = x or z ==> x by A2;
    thus z = y or z ==> y by A2;
  end;
