 reserve n for Nat;

theorem AxiomA8: :: Axiom A8
  ex a,b,c being Point of TarskiEuclid2Space st
    not between a,b,c & not between b,c,a & not between c,a,b
  proof
    the MetrStruct of TarskiEuclid2Space = the MetrStruct of Euclid 2
      by GTARSKI1:def 13;
    then reconsider a = |[0,0]|, b = |[1,0]|, c = |[0,1]|
      as Point of TarskiEuclid2Space by EUCLID:22;
    take a,b,c;
    thus not between a,b,c
    proof
      assume between a,b,c;
      then Tn2TR b in LSeg(Tn2TR a,Tn2TR c) by ThConv6;
      then dist(Tn2TR a,Tn2TR b) + dist(Tn2TR b,Tn2TR c) =
        dist(Tn2TR a,Tn2TR c) by EUCLID12:12;
      hence thesis by SQUARE_1:19,THY1,THY2,THY3;
    end;
    thus not between b,c,a
    proof
      assume between b,c,a;
      then Tn2TR c in LSeg(Tn2TR b,Tn2TR a) by ThConv6;
      then dist(Tn2TR b,Tn2TR c) + dist(Tn2TR c,Tn2TR a) =
        dist(Tn2TR b,Tn2TR a) by EUCLID12:12;
      hence thesis by SQUARE_1:19,THY1,THY2,THY3;
    end;
    assume between c,a,b;
    then Tn2TR a in LSeg(Tn2TR c,Tn2TR b) by ThConv6;
    then dist(Tn2TR c,Tn2TR a) + dist(Tn2TR a,Tn2TR b) =
      dist(Tn2TR c,Tn2TR b) by EUCLID12:12;
    hence thesis by SQUARE_1:21,THY1,THY2,THY3;
  end;
