
theorem Th17:
for seq be convergent ExtREAL_sequence holds
  (seq is convergent_to_finite_number iff -seq is convergent_to_finite_number)
& (seq is convergent_to_+infty iff -seq is convergent_to_-infty)
& (seq is convergent_to_-infty iff -seq is convergent_to_+infty)
& -seq is convergent & lim (-seq) = - lim seq
proof
   let seq be convergent ExtREAL_sequence;
   now assume -seq is convergent_to_finite_number; then
    -(-seq) is convergent_to_finite_number by Lm6;
    hence seq is convergent_to_finite_number by Th2;
   end;
   hence seq is convergent_to_finite_number
    iff -seq is convergent_to_finite_number by Lm6;
   now assume -seq is convergent_to_-infty; then
    -(-seq) is convergent_to_+infty by Lm6;
    hence seq is convergent_to_+infty by Th2;
   end;
   hence seq is convergent_to_+infty iff -seq is convergent_to_-infty by Lm6;
   now assume -seq is convergent_to_+infty; then
    -(-seq) is convergent_to_-infty by Lm6;
    hence seq is convergent_to_-infty by Th2;
   end;
   hence seq is convergent_to_-infty iff -seq is convergent_to_+infty by Lm6;
   thus -seq is convergent & lim (-seq) = - lim seq by Lm6;
end;
