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 UN iff the reduction of X is with_UN_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 UN implies R is with_UN_property
    proof
      assume
A1:   for x,y st x is normform & y is normform & x <=*=> y holds x = y;
      let a,b be object;
      assume
A2:   a is_a_normal_form_wrt R & b 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 & y is normform & x <=*=> y by A2,Ch1;
        hence a = b by A1;
      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;
        hence a = b by A2,REWRITE1:28,31;
      end;
    end;
    assume
A4: for a,b being object
    st a is_a_normal_form_wrt R & b is_a_normal_form_wrt R &
    a,b are_convertible_wrt R holds a = b;
    let x,y; assume
    x is normform & y is normform & x <=*=> y; then
    x is_a_normal_form_wrt R & y is_a_normal_form_wrt R &
    x,y are_convertible_wrt R by Ch1;
    hence x = y by A4;
  end;
