reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;
reserve r,r1,r2,g,g1,g2 for Real;

theorem Th81:
  f is convergent_in+infty implies -f is convergent_in+infty &
  lim_in+infty(-f)=-(lim_in+infty f)
proof
  assume
A1: f is convergent_in+infty;
  thus -f is convergent_in+infty by A1,Th80;
  thus lim_in+infty (-f)=(-jj)*(lim_in+infty f) by A1,Th80
    .=-(lim_in+infty f);
end;
