 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 ::: I1:
  a <> b & x <> y & a on_line x,y & b on_line x,y
    implies x,y equal_line a,b
   proof
     assume
H1:  a <> b & x <> y; then
P2:  b,a equal_line a,b by LineEqA1;
     assume
H2:  a on_line x,y & b on_line x,y;
     per cases;
     suppose x = b; then
       x,y equal_line b,a by H1, H2, I1part2;
       hence thesis by P2;
     end;
     suppose
       x <> b; then
P4:    x,y equal_line x,b by H2, I1part2; then
       x,b equal_line a,b by H1, I1part2Reverse, H2;
       hence thesis by P4;
     end;
  end;
