
theorem ThConv5:
  for p,q,r being POINT of TarskiEuclid2Space holds
    p is_Between q,r iff Tn2TR p in LSeg(Tn2TR q, Tn2TR r)
  proof
    let p,q,r be POINT of TarskiEuclid2Space;
A1: dist(Tn2TR q,Tn2TR p) = dist(Tn2E q,Tn2E p) &
    dist(Tn2TR p,Tn2TR r)=dist(Tn2E p,Tn2E r) & dist(Tn2TR q, Tn2TR r)
    = dist(Tn2E q,Tn2E r) by TOPREAL6:def 1;
    dist(q,p) = dist(Tn2E q,Tn2E p) & dist(p,r)=dist(Tn2E p,Tn2E r) &
      dist(q,r)=dist(Tn2E q,Tn2E r) by ThConv;
    hence thesis by A1,EUCLID12:12;
  end;
