
theorem Th39:
for X be non empty set, F be Functional_Sequence of X,ExtREAL,
 x be Element of X holds
  (F#x is convergent_to_+infty iff (-F)#x is convergent_to_-infty) &
  (F#x is convergent_to_-infty iff (-F)#x is convergent_to_+infty) &
  (F#x is convergent_to_finite_number
    iff (-F)#x is convergent_to_finite_number) &
  (F#x is convergent iff (-F)#x is convergent) &
  (F#x is convergent implies lim((-F)#x) = - lim(F#x))
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL,
    x be Element of X;
A1: F#x is convergent_to_+infty implies
     -(F#x) is convergent_to_-infty by DBLSEQ_3:17;
    -(F#x) is convergent_to_-infty implies
     -(-(F#x)) is convergent_to_+infty by DBLSEQ_3:17;
    hence A2: F#x is convergent_to_+infty
     iff (-F)#x is convergent_to_-infty by A1,DBLSEQ_3:2,Th38;
A3: F#x is convergent_to_-infty implies
     -(F#x) is convergent_to_+infty by DBLSEQ_3:17;
    -(F#x) is convergent_to_+infty implies
     -(-(F#x)) is convergent_to_-infty by DBLSEQ_3:17;
    hence A4: F#x is convergent_to_-infty
     iff (-F)#x is convergent_to_+infty by A3,DBLSEQ_3:2,Th38;
A5: F#x is convergent_to_finite_number implies
      -(F#x) is convergent_to_finite_number by DBLSEQ_3:17;
    -(F#x) is convergent_to_finite_number implies
      -(-(F#x)) is convergent_to_finite_number by DBLSEQ_3:17;
    hence A6: F#x is convergent_to_finite_number
      iff (-F)#x is convergent_to_finite_number by A5,Th38,DBLSEQ_3:2;
    thus F#x is convergent iff (-F)#x is convergent
      by A2,A4,A6,MESFUNC5:def 11;
    hereby assume F#x is convergent; then
     lim (-(F#x)) = - lim(F#x) by DBLSEQ_3:17;
     hence lim((-F)#x) = - lim(F#x) by Th38;
    end;
end;
