 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 BextendToLine:
  a <> b & between a,b,c & between a,b,d
    implies ex x st a,b,c,x are_ordered & a,b,d,x are_ordered
   proof
     assume that
H1:  a <> b and
H2:  between a,b,c and
H3:  between a,b,d;
     consider u such that
X1:  between a,c,u & c,u equiv b,d by A4;
X2:  a,b,c,u are_ordered by H2, X1, B123and134Ordered; then
X3:  between u,c,b by Bsymmetry;
     u,c equiv c,u by A1; then
X4:  u,c equiv b,d by X1, EquivTransitive;
     consider x such that
Y1:  between a,d,x & d,x equiv b,c by A4;
Y2:  a,b,d,x are_ordered by H3, Y1, B123and134Ordered;
Y4:  b,c equiv d,x by Y1, EquivSymmetric;
     c,b equiv b,c by A1; then
     c,b equiv d,x by Y4, EquivTransitive; then
X6:  u,b equiv b,x by X3, Y2, X4, SegmentAddition;
     b,u equiv u,b by A1; then
     u = x by H1, X2, Y2, C1, X6, EquivTransitive;
     hence thesis by X2, Y2;
   end;
