 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 EqDist2PointsBetween:
  a <> b & between a,b,c & a,p equiv a,q & b,p equiv b,q
    implies c,p equiv c,q
   proof
     assume
H1:  a <> b;
     assume
H2:  between a,b,c;
     assume
H3:  a,p equiv a,q & b,p equiv b,q;
::   a & b are equidistant from p & q.  Apply SAS to a+pbc & a+qbc.
     a,b,p cong a,b,q by H3, EquivReflexive; then
     p,c equiv q,c by H1, H2, EquivReflexive, A5;
     hence thesis by CongruenceDoubleSymmetry;
   end;
