Let a and t be given.
Assume HaR: a ∈ R.
Assume HtR: t ∈ R.
We will prove (apply_fun normalize01_fun t ∈ open_ray_upper R a ↔ Rlt a (apply_fun normalize01_fun t)).
Apply iffI to the current goal.
Assume Hray: apply_fun normalize01_fun t ∈ open_ray_upper R a.
We will prove Rlt a (apply_fun normalize01_fun t).
We prove the intermediate claim Hrel: order_rel R a (apply_fun normalize01_fun t).
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ order_rel R a x0) (apply_fun normalize01_fun t) Hray).
An exact proof term for the current goal is (order_rel_R_implies_Rlt a (apply_fun normalize01_fun t) Hrel).
Assume Halt: Rlt a (apply_fun normalize01_fun t).
We will prove apply_fun normalize01_fun t ∈ open_ray_upper R a.
We prove the intermediate claim HfR: apply_fun normalize01_fun t ∈ R.
An exact proof term for the current goal is (normalize01_fun_value_in_R t HtR).
We prove the intermediate claim Hrel: order_rel R a (apply_fun normalize01_fun t).
An exact proof term for the current goal is (Rlt_implies_order_rel_R a (apply_fun normalize01_fun t) Halt).
An exact proof term for the current goal is (SepI R (λx0 : set ⇒ order_rel R a x0) (apply_fun normalize01_fun t) HfR Hrel).
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