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
  X is WN iff the reduction of X is weakly-normalizing
  proof set R = the reduction of X;
A0: field R c= (the carrier of X)\/the carrier of X by RELSET_1:8;
    thus X is WN implies R is weakly-normalizing
    proof assume
A1:   for x holds x is normalizable;
      let a be object; assume a in field R; then
      reconsider a as Element of X by A0;
      a is normalizable by A1;
      hence thesis by Ch3;
    end;
    assume
A2: for a being object st a in field R
    holds a has_a_normal_form_wrt R;
    let x;
    per cases;
    suppose
      x in field R;
      hence thesis by A2,Ch3;
    end;
    suppose
A3:   not x in field R;
      take x;
      thus x is normform
      proof
        let y;
        thus not [x,y] in R by A3,RELAT_1:15;
      end;
      thus thesis;
    end;
  end;
