 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 FlatNormal:
  between d,e,d9 & c,d9 equiv c,d & d,e equiv d9,e &
    c <> d & e <> d implies
    ex p,r,q st between p,r,q & between r,c,d9 & between e,c,p &
      r,c,p cong r,c,q & r,c equiv e,c & p,r equiv d,e
   proof
     assume that
H2:  between d,e,d9 and
H3:  c,d9 equiv c,d and
H4:  d,e equiv d9,e and
H5:  c <> d and
H6:  e <> d;
     c <> d9 by H5, H3, EquivSymmetric, A3; then
     consider p,r such that
X1:  between e,c,p & between d9,c,r & p,r,c cong d9,e,c
       by EasyAngleTransport;
     d9,e equiv d,e by H4, EquivSymmetric; then
X3:  p,r equiv d,e by X1, EquivTransitive; then
X4:  p <> r by EquivSymmetric, H6, A3;
     consider q such that
X5:  between p,r,q & r,q equiv e,d by A4;
X6:  between d9,e,d by H2, Bsymmetry;
     c,p equiv c,d9 by X1, CongruenceDoubleSymmetry; then
X7:  c,p equiv c,d by H3, EquivTransitive;
::   Apply SAS to p+crq & d9+ced
     c,q equiv c,d by X4, X1, X5, X6, A5; then
     c,d equiv c,q by EquivSymmetric; then
X8:  c,p equiv c,q by X7, EquivTransitive;
X10:  r,p equiv e,d by X3, CongruenceDoubleSymmetry;
     e,d equiv r,q by X5, EquivSymmetric; then
X11: r,c,p cong r,c,q by EquivReflexive, X8, X10, EquivTransitive;
     between r,c,d9 by X1, Bsymmetry;
     hence thesis by X5, X11, X1, X3;
   end;
