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

theorem Th37:
  for R being Relation, a,b being object st a,b are_convergent_wrt R
  or a,b are_divergent_wrt R holds a,b are_convertible_wrt R
proof
  let R be Relation, a,b be object;
  assume
A1: a,b are_convergent_wrt R or a,b are_divergent_wrt R;
  per cases by A1;
  suppose
    a,b are_convergent_wrt R;
    then consider c being object such that
A2: R reduces a,c & R reduces b,c;
    a,c are_convertible_wrt R & c,b are_convertible_wrt R by A2,Th25;
    hence thesis by Th30;
  end;
  suppose
    a,b are_divergent_wrt R;
    then consider c being object such that
A3: R reduces c,a & R reduces c,b;
    c,b are_convertible_wrt R & a,c are_convertible_wrt R by A3,Th25;
    hence thesis by Th30;
  end;
end;
