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
  X is N.F. iff the reduction of X is with_NF_property
  proof
    set R = the reduction of X;
    set A = the carrier of X;
A0: field R c= A \/ A by RELSET_1:8;
    thus X is N.F. implies R is with_NF_property
    proof
      assume
A1:   for x,y st x is normform & x <=*=> y holds y =*=> x;
      let a,b be object; assume
A2:   a is_a_normal_form_wrt R & a,b are_convertible_wrt R;
      per cases;
      suppose a in A & b in A; then
        reconsider x = a, y = b as Element of X;
        x is normform & x <=*=> y by A2,Ch1; then
        y =*=> x by A1;
        hence R reduces b,a;
      end;
      suppose not a in A or not b in A; then
        not a in field R or not b in field R by A0; then
        a = b by A2,REWRITE1:28,31;
        hence R reduces b,a by REWRITE1:12;
      end;
    end;
    assume
B1: for a,b being object st
      a is_a_normal_form_wrt R & a,b are_convertible_wrt R
    holds R reduces b,a;
    let x,y; assume
    x is normform & x <=*=> y;
    hence R reduces y,x by B1,Ch1;
  end;
