theorem
  a,b,c are_collinear & a,c,d are_collinear & a<>c implies b,c,d are_collinear
proof
  assume that
A1: a,b,c are_collinear and
A2: a,c,d are_collinear & a<>c;
A3: a,c,c are_collinear by Th12;
  a,c,b are_collinear by A1,Th10;
  hence thesis by A2,A3,Th13;
end;
