 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 I1part1:
  a <> b & a <> x & x on_line a,b &
    c on_line a,b implies c on_line a,x
   proof
     assume that
H1:  a <> b and
H2:  a <> x;
     assume x on_line a,b; then
X1:  between a,b,x or between a,x,b or between x,a,b by Bsymmetry;
     assume
H4:  c on_line a,b;
     between a,x,c or between a,c,x or between x,a,c
     proof
       consider m being POINT of S such that
X5:    between b,a,m & a,m equiv a,b by A4;
X6:    a <> m by H1, X5, EquivSymmetric, A3;
       per cases by X1, Bsymmetry, H4;
       suppose
X3:      between a,b,x & between a,b,c; then
         between b,x,c or between b,c,x by H1, Gupta; then
         a,b,x,c are_ordered or a,b,c,x are_ordered
           by X3, BTransitivityOrdered;
         hence thesis;
       end;
       suppose
         between a,b,x & between a,c,b; then
         a,c,b,x are_ordered by B123and134Ordered;
         hence thesis;
       end;
       suppose
         between a,b,x & between c,a,b; then
         c,a,b,x are_ordered by H1, BTransitivityOrdered;
         hence thesis by Bsymmetry;
       end;
       suppose
         between a,x,b & between a,b,c; then
         a,x,b,c are_ordered by B123and134Ordered;
         hence thesis;
       end;
       suppose
X4:      between a,x,b & between a,c,b;
         between m,a,b by X5, Bsymmetry; then  :: m,a,c,b
         between m,a,c & between m,a,x by X4, B124and234then123;
         hence thesis by X6, Gupta;
       end;
       suppose
         between a,x,b & between c,a,b; then  :: c,a,x,b
         between c,a,x by B124and234then123;
         hence thesis by Bsymmetry;
       end;
       suppose
         between x,a,b & between a,b,c; then
         x,a,b,c are_ordered by H1, BTransitivityOrdered;
         hence thesis;
       end;
       suppose
         between x,a,b & between a,c,b; :: x,a,c,b
         hence thesis by B124and234then123;
       end;
       suppose
         between x,a,b & between c,a,b; then
         between b,a,x & between b,a,c by Bsymmetry;
         hence thesis by H1, Gupta;
       end;
     end;
     hence thesis by H2, Bsymmetry;
   end;
