 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 EqDist2PointsInnerBetween:
  between a,x,c & a,p equiv a,q & c,p equiv c,q
    implies x,p equiv x,q
   proof
     assume
H1:  between a,x,c;
     assume
H2:  a,p equiv a,q & c,p equiv c,q;
::   a and c are equidistant from p and q.  Apply Inner5Segments to apb-x & aqb-x.
X1:  a,c equiv a,c & c,x equiv c,x by EquivReflexive; then
     a,p,c cong a,q,c by H2, CongruenceDoubleSymmetry; then
     p,x equiv q,x by H1, X1, Inner5Segments;
     hence thesis by CongruenceDoubleSymmetry;
   end;
