 reserve S for satisfying_Tarski-model TarskiGeometryStruct;
 reserve a, b, c, d, e, f, o, p, q, r, s,
    v, w, u, x, y, z, a9, b9, c9, d9, x9, y9, z for POINT of S;

theorem BTransitivityOrdered:
  b <> c & between a,b,c & between b,c,d implies a,b,c,d are_ordered
   proof
     assume that
H1:  b <> c and
H2:  between a,b,c;
     assume
H3:  between b,c,d; then
X1:  between a,c,d by H1, H2, BTransitivity;
X2:  between d,c,b by H3, Bsymmetry;
     between c,b,a by H2, Bsymmetry; then
     between d,b,a by H1, X2, BTransitivity;
     hence thesis by H2, X1, H3, Bsymmetry;
   end;
