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 Ch3:
  x is normalizable iff x has_a_normal_form_wrt the reduction of X
  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 normalizable implies x has_a_normal_form_wrt R
    proof
      given y such that
A1:   y is_normform_of x;
      take y; thus thesis by A1,Ch2;
    end;
    given a being object such that
A2: a is_a_normal_form_of x, R;
    R reduces x,a by A2,REWRITE1:def 6; then
    x = a or a in field R by REWRITE1:18; then
    reconsider a as Element of X by A0;
    take a; thus thesis by A2,Ch2;
  end;
