 reserve n for Nat;

theorem AxiomA9: :: Axiom A9
  for a,b,c,p,q being Point of TarskiEuclid2Space st
    p <> q & a,p equiv a,q & b,p equiv b,q & c,p equiv c,q holds
      between a, b, c or between b, c, a or between c, a, b
  proof
    let a,b,c,p,q be Point of TarskiEuclid2Space;
    assume that
A1: p <> q and
A2: a,p equiv a,q and
A3: b,p equiv b,q and
A4: c,p equiv c,q;
    |. Tn2TR a - Tn2TR p.| = |. Tn2TR a - Tn2TR q.| by A2,ThConv4; then
A5: Tn2TR a in the_perpendicular_bisector(Tn2TR p,Tn2TR q) by A1,EUCLID12:60;
    |. Tn2TR b - Tn2TR p.| = |. Tn2TR b - Tn2TR q.| by A3,ThConv4; then
A6: Tn2TR b in the_perpendicular_bisector(Tn2TR p,Tn2TR q) by A1,EUCLID12:60;
    |. Tn2TR c - Tn2TR p.| = |. Tn2TR c - Tn2TR q.| by A4,ThConv4;
    then Tn2TR c in the_perpendicular_bisector(Tn2TR p,Tn2TR q)
      by A1,EUCLID12:60;
    then Tn2TR a in LSeg(Tn2TR b,Tn2TR c) or
      Tn2TR b in LSeg(Tn2TR c,Tn2TR a) or Tn2TR c in LSeg(Tn2TR a,Tn2TR b)
      by A5,A6,ThAZ9;
    hence thesis by ThConv6;
  end;
