A triangle inequality for angles in a Hilbert space

Let  x,y,z be unit vectors in a Hilbert space, and define the angle θ_xy by cos θ_xy  = Re (x,y), 0 ≤ θ_xy ≤ π.  The object this note  is to give a proof of the following inequality

(1)                   θ_xz  < θ_xy  + θ_yz .