Section Addition.
Variable (m n p: nat) (x y z: nat).
Hypothesis (Ex: encodes (sigma (F x)) m) (Ey: encodes (sigma (F y)) n) (Ez: encodes (sigma (F z)) p).

Lemma backward_add:
sigma • fst (addEQ x y z) ≡ sigma • snd (addEQ x y z) -> m + n = p.