reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;
reserve x,y,z for set, i,j for Nat;

theorem Th18:
  for R being weakly-normalizing with_UN_property Relation
  holds nf(nf(x,R),R) = nf(x,R)
  proof
    let R be weakly-normalizing with_UN_property Relation;
    nf(x,R) is_a_normal_form_of x,R by REWRITE1:54;
    then
A1: nf(x,R) is_a_normal_form_wrt R & R reduces nf(x,R),nf(x,R)
    by REWRITE1:12;
    then
A2: nf(x,R) is_a_normal_form_of nf(x,R),R;
    nf(x,R) has_a_normal_form_wrt R &
    for b,c being object st b is_a_normal_form_of nf(x,R),R & c
    is_a_normal_form_of nf(x,R),R holds b = c
    by A1,REWRITE1:def 6,53;
    hence nf(nf(x,R),R) = nf(x,R) by A2,REWRITE1:def 12;
  end;
