reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th3:
  r in rng seq iff -r in rng(-seq)
proof
  thus r in rng seq implies -r in rng(-seq)
  proof
    assume r in rng seq;
    then consider n being Element of NAT such that
A1: r=seq.n by FUNCT_2:113;
    -r=(-seq).n by A1,SEQ_1:10;
    hence thesis by VALUED_0:28;
  end;
  assume -r in rng(-seq);
  then consider n being Element of NAT such that
A2: -r=(-seq).n by FUNCT_2:113;
  r=-(-seq).n by A2;
  then r= - -seq.n by SEQ_1:10;
  hence thesis by VALUED_0:28;
end;
