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;
