 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 RhombusDiagBisect:
  between b,c,d9 & between b,d,c9 &
    c,d9 equiv c,d & d,c9 equiv c,d & d9,c9 equiv c,d implies
      ex e st between c,e,c9 & between d,e,d9 &
    c,e equiv c9,e & d,e equiv d9,e
   proof
     assume that
H1:  between b,c,d9 and
H2:  between b,d,c9 and
H3:  c,d9 equiv c,d and
H4:  d,c9 equiv c,d and
H5:  d9,c9 equiv c,d;
     between d9,c,b & between c9,d,b by H1, H2, Bsymmetry;
     then consider e such that
X2:  between c,e,c9 & between d,e,d9 by A7;
X3:  c,d equiv c,d9 by H3, EquivSymmetric;
X4:  c,c9 equiv c,c9 by EquivReflexive;
     c,d equiv d9,c9 by H5, EquivSymmetric; then
     c,d,c9 cong c,d9,c9 by X3, X4, H4, EquivTransitive; then
X5:  d,e equiv d9,e by X2, EquivReflexive, Inner5Segments;
X6:  d,c equiv c,d by A1;
     c,d equiv d,c9 by H4, EquivSymmetric; then
X7:  d,c equiv d,c9 by X6, EquivTransitive;
X9:  c,d equiv d9,c9 by H5, EquivSymmetric;
     d9,c9 equiv c9,d9 by A1; then
     c,d equiv c9,d9 by X9, EquivTransitive; then
     c,d9 equiv c9,d9 by H3, EquivTransitive; then
     d,c,d9 cong d,c9,d9 by X7, EquivReflexive; then
     c,e equiv c9,e by X2, EquivReflexive, Inner5Segments;
     hence thesis by X2, X5;
   end;
