reserve p,q,r for FinSequence,
  x,y for object;

theorem Th55:
  for R being with_UN_property weakly-normalizing Relation
  for a,b being object st a,b are_convertible_wrt R holds
    nf(a,R) = nf(b,R)
proof
  let R be with_UN_property weakly-normalizing Relation;
  let a,b be object;
A1: nf(b,R) is_a_normal_form_of b,R by Th54;
  then
A2: nf(b,R) is_a_normal_form_wrt R;
  R reduces b,nf(b,R) by A1;
  then
A3: b,nf(b,R) are_convertible_wrt R by Th25;
A4: nf(a,R) is_a_normal_form_of a,R by Th54;
  then R reduces a,nf(a,R);
  then
A5: nf(a,R),a are_convertible_wrt R by Th25;
  assume a,b are_convertible_wrt R;
  then nf(a,R),b are_convertible_wrt R by A5,Th30;
  then
A6: nf(a,R),nf(b,R) are_convertible_wrt R by A3,Th30;
  nf(a,R) is_a_normal_form_wrt R by A4;
  hence thesis by A2,A6,Def19;
end;
