We will prove one_third < two_thirds.
We prove the intermediate claim H13R: one_third ∈ R.
An exact proof term for the current goal is one_third_in_R.
We prove the intermediate claim H13S: SNo one_third.
An exact proof term for the current goal is (real_SNo one_third H13R).
We prove the intermediate claim Hlt13: 0 < one_third.
An exact proof term for the current goal is one_third_pos.
We prove the intermediate claim Hdef: two_thirds = add_SNo one_third one_third.
Use reflexivity.
rewrite the current goal using Hdef (from left to right).
rewrite the current goal using (add_SNo_0R one_third H13S) (from right to left) at position 1.
An exact proof term for the current goal is (add_SNo_Lt2 one_third 0 one_third H13S SNo_0 H13S Hlt13).
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