theorem Th32:
  f is_convergent_in x0 implies -f is_convergent_in x0 & lim(-f,x0
  )=-(lim(f,x0))
proof
  assume
A1: f is_convergent_in x0;
  thus -f is_convergent_in x0 by A1,Th31;
  thus lim(-f,x0)=(-1)*(lim(f,x0)) by A1,Th31
    .=-(lim(f,x0));
end;
