 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 C1prime:
  a <> b & between a,b,x & between a,b,y & a,x equiv a,y
    implies x = y
  proof
    assume that
H1: a <> b and
H2: between a,b,x and
H3: between a,b,y and
H4: a,x equiv a,y;
    consider m being POINT of S such that
X1: between b,a,m & a,m equiv a,b by A4;
X3: m <> a by X1, EquivSymmetric, A3, H1;
X2: between m,a,b by X1, Bsymmetry; then
x4: m,a,b,x are_ordered by H1, H2, BTransitivityOrdered;
    m,a,b,y are_ordered by H1, X2, H3, BTransitivityOrdered;
    hence x = y by X3, x4, H4, C1;
  end;
