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 Th21:
  (for n holds seq.n=-n) implies seq is divergent_to-infty
proof
  assume
A1: for n holds seq.n=-n;
A2: now
    let n be Element of NAT;
    thus (-s1).n=-s1.n by SEQ_1:10
      .=-n
      .=seq.n by A1;
  end;
  s1 is divergent_to+infty by Lm5,Th20;
  then (-jj)(#)s1 is divergent_to-infty by Th13;
  hence thesis by A2,FUNCT_2:63;
end;
