 reserve n for Nat;
 reserve X,Y for Subset of TarskiEuclid2Space;

theorem
  for a being Element of TarskiEuclid2Space st
    (for x,y being Element of TarskiEuclid2Space st x in X & y in Y holds
      between a,x,y) & a in Y holds X = {a} or X is empty
  proof
    let a be Element of TarskiEuclid2Space;
    assume that
A1: for x,y being Element of TarskiEuclid2Space st
      x in X & y in Y holds between a,x,y and
A2: a in Y;
M1: X c= {a}
    proof
      let x be object;
      assume
  L1: x in X;
      then reconsider x as Element of TarskiEuclid2Space;
      a = x by GTARSKI1:def 10,A1,A2,L1;
      hence thesis by TARSKI:def 1;
    end;
    per cases;
    suppose X is empty;
      hence thesis;
    end;
    suppose X is non empty;
      then consider x be object such that A3: x in X;
      reconsider x as Element of TarskiEuclid2Space by A3;
      a = x by GTARSKI1:def 10,A1,A2,A3;
      then {a} c= X by TARSKI:def 1,A3;
      hence thesis by M1;
    end;
  end;
