 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 ::: B124and234Ordered:
  between a,b,d & between b,c,d implies a,b,c,d are_ordered
   proof
     assume that
H1:  between a,b,d and
H2:  between b,c,d;
     per cases;
     suppose b = c;
       hence a,b,c,d are_ordered by H1, H2, Bqaa;
     end;
     suppose
Q1:    b <> c;
       between a,b,c by H1, H2, B124and234then123;
       hence a,b,c,d are_ordered by Q1, H2, BTransitivityOrdered;
     end;
   end;
