theorem Th10:
  not a,b,c are_collinear & a,b,d are_collinear & c,e,d are_collinear
  & e<>c & d<>a implies not e,a,c are_collinear
proof
  assume that
A1: not a,b,c are_collinear and
A2: a,b,d are_collinear and
A3: c,e,d are_collinear & e<>c and
A4: d<>a;
  assume not thesis;
  then c,e,a are_collinear by Th1;
  then c,a,d are_collinear by A3,Th2;
  then
A5: a,d,c are_collinear by Th1;
  a,d,b are_collinear by A2,Th1;
  hence contradiction by A1,A4,A5,Th2;
end;
