 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 BTransitivity:
  b <> c & between a,b,c & between b,c,d implies between a,c,d
  proof
     assume that
H1:  b <> c and
H2:  between a,b,c and
H3:  between b,c,d;
     consider x such that
X1:  between a,c,x & c,x equiv c,d by A4;
X2:  between x,c,a by X1, Bsymmetry;
     between c,b,a by H2, Bsymmetry; then
     between x,c,b by X2, B124and234then123; then
     between b,c,x by Bsymmetry;
     hence thesis by X1, H1, H3, C1;
  end;
