 reserve n for Nat;

theorem THNOIX3:
  for p,q,r,s being Point of TarskiEuclid2Space st
    q <> r & between p,q,r & between q,r,s holds between p,q,s
  proof
    let p,q,r,s be Point of TarskiEuclid2Space;
    assume that
A1: q <> r and
A2: between p,q,r and
A3: between q,r,s;
A4: Tn2TR q in LSeg(Tn2TR p,Tn2TR r) by A2,ThConv6; then
A5: dist(Tn2TR p, Tn2TR q)+dist(Tn2TR q, Tn2TR r)=
      dist(Tn2TR p, Tn2TR r) by EUCLID12:12;
A6: Tn2TR r in LSeg(Tn2TR q,Tn2TR s) by A3,ThConv6;
A7: Tn2TR r in LSeg(Tn2TR p, Tn2TR s) by A1,A4,A6,THORANGE2;
    dist(Tn2TR p, Tn2TR q) + dist(Tn2TR q, Tn2TR s)
      = dist(Tn2TR p, Tn2TR r) - dist(Tn2TR q, Tn2TR r) +
        (dist(Tn2TR q, Tn2TR r) + dist(Tn2TR r, Tn2TR s))
        by A5,A6,EUCLID12:12
     .= dist(Tn2TR p, Tn2TR r) + dist(Tn2TR r, Tn2TR s)
     .= dist(Tn2TR p, Tn2TR s) by A7,EUCLID12:12;
    then Tn2TR q in LSeg(Tn2TR p, Tn2TR s) by EUCLID12:12;
    hence thesis by ThConv6;
  end;
