Let x, y, z, w, u and v be given.

Assume Hx Hy Hz Hw Hu Hv.

Assume H1: x + y + v < w + u + z.

We will prove x + y + - z < w + u + - v.

We prove the intermediate claim Lmv: SNo (- v).

An exact proof term for the current goal is SNo_minus_SNo v Hv.

Apply add_SNo_minus_Lt1b3 x y z (w + u + - v) Hx Hy Hz (SNo_add_SNo_3 w u (- v) Hw Hu Lmv) to the current goal.

We will prove x + y < (w + u + - v) + z.

rewrite the current goal using add_SNo_assoc w u (- v) Hw Hu Lmv (from left to right).

We will prove x + y < ((w + u) + - v) + z.

rewrite the current goal using add_SNo_com_3b_1_2 (w + u) (- v) z (SNo_add_SNo w u Hw Hu) Lmv Hz (from left to right).

We will prove x + y < ((w + u) + z) + - v.

Apply add_SNo_minus_Lt2b ((w + u) + z) v (x + y) (SNo_add_SNo (w + u) z (SNo_add_SNo w u Hw Hu) Hz) Hv (SNo_add_SNo x y Hx Hy) to the current goal.

We will prove (x + y) + v < (w + u) + z.

rewrite the current goal using add_SNo_assoc x y v Hx Hy Hv (from right to left).

rewrite the current goal using add_SNo_assoc w u z Hw Hu Hz (from right to left).

An exact proof term for the current goal is H1.

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