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 Ch14:
  x >>01<< y iff x,y are_convergent<=1_wrt the reduction of X
  proof set R = the reduction of X;
    thus x >>01<< y implies x,y are_convergent<=1_wrt R
    proof
      given z such that
A1:   z <=01= x & y =01=> z;
      take z;
      (x ==> z or z = x) & (y ==> z or z = y) by A1;
      hence ([x,z] in R or x = z) & ([y,z] in R or y = z);
    end;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    given a being object such that
A2: ([x,a] in R or x = a) & ([y,a] in R or y = a);
    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 x = z or x ==> z by A2;
    thus y = z or y ==> z by A2;
  end;
