We will prove continuous_map R R_standard_topology R R_standard_topology neg_fun.
Set Tx to be the term R_standard_topology.
Set S to be the term open_rays_subbasis R.
We prove the intermediate claim HTx: topology_on R Tx.
An exact proof term for the current goal is R_standard_topology_is_topology.
We prove the intermediate claim Hfun: function_on neg_fun R R.
Let t be given.
Assume HtR: t ∈ R.
An exact proof term for the current goal is (neg_fun_value_in_R t HtR).
We prove the intermediate claim HS: subbasis_on R S.
An exact proof term for the current goal is (open_rays_subbasis_is_subbasis R).
We prove the intermediate claim Hgen: generated_topology_from_subbasis R S = Tx.
rewrite the current goal using open_rays_subbasis_for_order_topology_R (from left to right).
rewrite the current goal using standard_topology_is_order_topology (from left to right).
Use reflexivity.
rewrite the current goal using Hgen (from right to left) at position 2.
We prove the intermediate claim HpreS: ∀s : set, s ∈ S → preimage_of R neg_fun s ∈ Tx.
Let s be given.
Assume HsS: s ∈ S.
We will prove preimage_of R neg_fun s ∈ Tx.
Apply (binunionE' ({I ∈ 𝒫 R|∃a ∈ R, I = open_ray_upper R a} ∪ {I ∈ 𝒫 R|∃b ∈ R, I = open_ray_lower R b}) {R} s (preimage_of R neg_fun s ∈ Tx)) to the current goal.
Assume Hs0: s ∈ ({I ∈ 𝒫 R|∃a ∈ R, I = open_ray_upper R a} ∪ {I ∈ 𝒫 R|∃b ∈ R, I = open_ray_lower R b}).
Apply (binunionE' {I ∈ 𝒫 R|∃a ∈ R, I = open_ray_upper R a} {I ∈ 𝒫 R|∃b ∈ R, I = open_ray_lower R b} s (preimage_of R neg_fun s ∈ Tx)) to the current goal.
Assume Hsu: s ∈ {I ∈ 𝒫 R|∃a ∈ R, I = open_ray_upper R a}.
We prove the intermediate claim Hex: ∃a ∈ R, s = open_ray_upper R a.
An exact proof term for the current goal is (SepE2 (𝒫 R) (λI0 : set ⇒ ∃a ∈ R, I0 = open_ray_upper R a) s Hsu).
Apply Hex to the current goal.
Let a be given.
Assume Hacore.
Apply Hacore to the current goal.
Assume HaR: a ∈ R.
Assume Hseq: s = open_ray_upper R a.
rewrite the current goal using Hseq (from left to right).
rewrite the current goal using (preimage_neg_fun_open_ray_upper a HaR) (from left to right).
rewrite the current goal using standard_topology_is_order_topology (from right to left).
We prove the intermediate claim HmaR: minus_SNo a ∈ R.
An exact proof term for the current goal is (real_minus_SNo a HaR).
We prove the intermediate claim HsRay: open_ray_lower R (minus_SNo a) ∈ open_rays_subbasis R.
An exact proof term for the current goal is (open_ray_lower_in_open_rays_subbasis R (minus_SNo a) HmaR).
An exact proof term for the current goal is (open_rays_subbasis_sub_order_topology_R (open_ray_lower R (minus_SNo a)) HsRay).
Assume Hsl: s ∈ {I ∈ 𝒫 R|∃b ∈ R, I = open_ray_lower R b}.
We prove the intermediate claim Hex: ∃b ∈ R, s = open_ray_lower R b.
An exact proof term for the current goal is (SepE2 (𝒫 R) (λI0 : set ⇒ ∃b ∈ R, I0 = open_ray_lower R b) s Hsl).
Apply Hex to the current goal.
Let b be given.
Assume Hbcore.
Apply Hbcore to the current goal.
Assume HbR: b ∈ R.
Assume Hseq: s = open_ray_lower R b.
rewrite the current goal using Hseq (from left to right).
rewrite the current goal using (preimage_neg_fun_open_ray_lower b HbR) (from left to right).
rewrite the current goal using standard_topology_is_order_topology (from right to left).
We prove the intermediate claim HmbR: minus_SNo b ∈ R.
An exact proof term for the current goal is (real_minus_SNo b HbR).
We prove the intermediate claim HsRay: open_ray_upper R (minus_SNo b) ∈ open_rays_subbasis R.
An exact proof term for the current goal is (open_ray_upper_in_open_rays_subbasis R (minus_SNo b) HmbR).
An exact proof term for the current goal is (open_rays_subbasis_sub_order_topology_R (open_ray_upper R (minus_SNo b)) HsRay).
An exact proof term for the current goal is Hs0.
Assume HsR: s ∈ {R}.
We prove the intermediate claim Hseq: s = R.
An exact proof term for the current goal is (SingE R s HsR).
rewrite the current goal using Hseq (from left to right).
We prove the intermediate claim Heq: preimage_of R neg_fun R = R.
An exact proof term for the current goal is (preimage_of_whole R R neg_fun Hfun).
rewrite the current goal using Heq (from left to right).
An exact proof term for the current goal is (topology_has_X R Tx HTx).
An exact proof term for the current goal is HsS.
An exact proof term for the current goal is (continuous_map_from_subbasis R Tx R S neg_fun HTx Hfun HS HpreS).
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