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 Th4:
  rng (-seq) = -- rng seq
proof
  thus rng (-seq) c= -- rng seq
  proof
    let x be object;
    assume
A1: x in rng(-seq);
    then reconsider r = x as Real;
    -r in rng -(-seq) by A1,Th3;
    then - -r in -- rng seq by MEASURE6:40;
    hence thesis;
  end;
  let x be object;
  assume
A2: x in -- rng seq;
  then reconsider r = x as Real;
  -r in -- -- rng seq by A2,MEMBER_1:12;
  then - -r in rng (-seq) by Th3;
  hence thesis;
end;
