reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem
  (for r be Point of S holds r in Y iff r in the carrier of S) iff Y=the
  carrier of S
proof
  thus (for r be Point of S holds r in Y iff r in the carrier of S) implies Y=
  the carrier of S
  proof
    assume for r be Point of S holds r in Y iff r in the carrier of S;
    then for y being object holds y in Y iff y in the carrier of S;
    hence thesis by TARSKI:2;
  end;
  assume
A1: Y=the carrier of S;
  let r be Point of S;
  thus r in Y implies r in the carrier of S;
  thus thesis by A1;
end;
