theorem Th15:
  not a,b,c,o are_coplanar & o,a,a9 are_collinear & a<>a9 implies
  not a,b,c,a9 are_coplanar
proof
  assume that
A1: not a,b,c,o are_coplanar and
A2: o,a,a9 are_collinear and
A3: a<>a9;
  assume
A4: not thesis;
A5: a,b,c,a are_coplanar by Th14;
  a,a9,o are_collinear by A2,Th1;
  hence contradiction by A1,A3,A4,A5,Th10;
end;
