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

theorem Th39:
  for a,b being object st a,b are_convergent_wrt {} or
  a,b are_divergent_wrt {} holds a = b
proof
  let a,b be object;
A1: now
    assume a,b are_convergent_wrt {};
    then consider c being object such that
A2: {} reduces a,c and
A3: {} reduces b,c;
    a = c by A2,Th13;
    hence thesis by A3,Th13;
  end;
A4: now
    assume a,b are_divergent_wrt {};
    then consider c being object such that
A5: {} reduces c,a and
A6: {} reduces c,b;
    a = c by A5,Th13;
    hence thesis by A6,Th13;
  end;
  assume a,b are_convergent_wrt {} or a,b are_divergent_wrt {};
  hence thesis by A1,A4;
end;
