Let x be given.
Assume Hx H1.
Let k be given.
Assume Hk.
Apply H1 k Hk to the current goal.
Let q be given.
Assume Hq.
Apply Hq to the current goal.
Assume Hq.
Apply Hq to the current goal.
Assume Hq1a Hq1b Hq1c Hq1d.
We prove the intermediate
claim Lqk:
SNo (q + eps_ k).
We use
(- (q + eps_ k)) to
witness the existential quantifier.
Apply andI to the current goal.
Apply andI to the current goal.
An exact proof term for the current goal is Hq3.
An exact proof term for the current goal is Hq2.
∎