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 Ch1:
  x is normform iff x is_a_normal_form_wrt the reduction of X
  proof set R = the reduction of X;
    thus x is normform implies x is_a_normal_form_wrt the reduction of X
    proof assume
Z0:   not ex y st x ==> y;
      let a be object;
      assume
Z1:   [x,a] in the reduction of X; then
      reconsider y = a as Element of X by ZFMISC_1:87;
      x ==> y by Z1;
      hence thesis by Z0;
    end;
    assume
Z1: not ex b being object st [x,b] in R;
    let y;
    assume [x,y] in the reduction of X;
    hence thesis by Z1;
  end;
