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
  (ex y st y is_normform_of x) &
  (for y,z st y is_normform_of x & z is_normform_of x holds y = z)
  implies nf x = nf(x, the reduction of X)
  proof set R = the reduction of X;
    set A = the carrier of X;
F0: field R c= A \/ A by RELSET_1:8;
    given y such that
A0: y is_normform_of x;
B0: x has_a_normal_form_wrt R by A0,Ch2,REWRITE1:def 11;
    assume
A1: for y,z st y is_normform_of x & z is_normform_of x holds y = z; then
    nf x is_normform_of x by A0,Def17; then
A2: nf x is_a_normal_form_of x,R by Ch2;
    now
      let b,c be object; assume
A3:   b is_a_normal_form_of x,R & c is_a_normal_form_of x,R; then
A4:   R reduces x,b & R reduces x,c by REWRITE1:def 6;
      per cases;
      suppose x in field R; then
        b in field R & c in field R by A4,REWRITE1:19; then
        reconsider y = b, z = c as Element of X by F0;
        y is_normform_of x & z is_normform_of x by A3,Ch2;
        hence b = c by A1;
      end;
      suppose not x in field R; then
        x = b & x = c by A4,REWRITE1:18;
        hence b = c;
      end;
    end;
    hence nf x = nf(x, the reduction of X) by B0,A2,REWRITE1:def 12;
 end;
