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

theorem
  for R being Relation, C being Completion of R, a,b being object holds
  a,b are_convertible_wrt R iff nf(a,C) = nf(b,C)
proof
  let R be Relation, C be Completion of R, a,b be object;
  nf(a,C) is_a_normal_form_of a,C by Th54;
  then
A1: C reduces a,nf(a,C);
  a,b are_convergent_wrt C implies a,b are_convertible_wrt C by Th37;
  hence a,b are_convertible_wrt R implies nf(a,C) = nf(b,C) by Def28,Th55;
  nf(b,C) is_a_normal_form_of b,C by Th54;
  then
A2: C reduces b,nf(b,C);
  a,b are_convertible_wrt R iff a,b are_convergent_wrt C by Def28;
  hence thesis by A1,A2;
end;
