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 Ch12:
  x >><< y iff x,y are_convergent_wrt the reduction of X
  proof set R = the reduction of X;
    thus x >><< y implies x,y are_convergent_wrt R
    proof
      given z such that
A1:   z <=*= x & y =*=> z;
      take z;
      thus R reduces x,z & R reduces y,z by A1;
    end;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    given a being object such that
A2: R reduces x,a & R reduces y,a;
    per cases;
    suppose
      a in field R; then
      reconsider z = a as Element of X by F0;
      take z;
      thus R reduces x,z & R reduces y,z by A2;
    end;
    suppose
      not a in field R; then
      a = x & a = y by A2,REWRITE1:18;
      hence thesis;
    end;
  end;
