 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 I1part2:
  a <> b & a <> x & x on_line a,b implies
    a,b equal_line a,x
   proof
     assume
H1:  a <> b & a <> x & x on_line a,b;
     c on_line a,b iff c on_line a,x
     proof
       thus c on_line a,b implies c on_line a,x by H1, I1part1;
       assume
H2:    c on_line a,x;
       b on_line a,x by H1, Bsymmetry;
       hence c on_line a,b by H1, H2, I1part1;
     end;
     hence thesis by H1;
   end;
