Let c and a be given.
Assume HcR: c ∈ R.
Assume Hcpos: 0 < c.
Assume HaR: a ∈ R.
We will prove preimage_of R (div_const_fun c) (open_ray_upper R a) = open_ray_upper R (mul_SNo a c).
Apply set_ext to the current goal.
Let t be given.
Assume HtPre: t ∈ preimage_of R (div_const_fun c) (open_ray_upper R a).
We will prove t ∈ open_ray_upper R (mul_SNo a c).
We prove the intermediate claim HtR: t ∈ R.
An exact proof term for the current goal is (SepE1 R (λx0 : set ⇒ apply_fun (div_const_fun c) x0 ∈ open_ray_upper R a) t HtPre).
We prove the intermediate claim Himg: apply_fun (div_const_fun c) t ∈ open_ray_upper R a.
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ apply_fun (div_const_fun c) x0 ∈ open_ray_upper R a) t HtPre).
We prove the intermediate claim Hrel: order_rel R a (apply_fun (div_const_fun c) t).
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ order_rel R a x0) (apply_fun (div_const_fun c) t) Himg).
We prove the intermediate claim Hrlt: Rlt a (apply_fun (div_const_fun c) t).
An exact proof term for the current goal is (order_rel_R_implies_Rlt a (apply_fun (div_const_fun c) t) Hrel).
We prove the intermediate claim HdivEq: apply_fun (div_const_fun c) t = div_SNo t c.
An exact proof term for the current goal is (div_const_fun_apply c t HcR HtR).
We prove the intermediate claim HtS: SNo t.
An exact proof term for the current goal is (real_SNo t HtR).
We prove the intermediate claim HcS: SNo c.
An exact proof term for the current goal is (real_SNo c HcR).
We prove the intermediate claim HaS: SNo a.
An exact proof term for the current goal is (real_SNo a HaR).
We prove the intermediate claim HacR: mul_SNo a c ∈ R.
An exact proof term for the current goal is (real_mul_SNo a HaR c HcR).
We prove the intermediate claim HacS: SNo (mul_SNo a c).
An exact proof term for the current goal is (real_SNo (mul_SNo a c) HacR).
We prove the intermediate claim Halt: a < div_SNo t c.
rewrite the current goal using HdivEq (from right to left).
An exact proof term for the current goal is (RltE_lt a (apply_fun (div_const_fun c) t) Hrlt).
We prove the intermediate claim HmulLt: mul_SNo a c < t.
An exact proof term for the current goal is (div_SNo_pos_LtR' t c a HtS HcS HaS Hcpos Halt).
We prove the intermediate claim Hray: order_rel R (mul_SNo a c) t.
An exact proof term for the current goal is (Rlt_implies_order_rel_R (mul_SNo a c) t (RltI (mul_SNo a c) t HacR HtR HmulLt)).
An exact proof term for the current goal is (SepI R (λx0 : set ⇒ order_rel R (mul_SNo a c) x0) t HtR Hray).
Let t be given.
Assume HtRay: t ∈ open_ray_upper R (mul_SNo a c).
We will prove t ∈ preimage_of R (div_const_fun c) (open_ray_upper R a).
We prove the intermediate claim HtR: t ∈ R.
An exact proof term for the current goal is (SepE1 R (λx0 : set ⇒ order_rel R (mul_SNo a c) x0) t HtRay).
We prove the intermediate claim Hrel: order_rel R (mul_SNo a c) t.
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ order_rel R (mul_SNo a c) x0) t HtRay).
We prove the intermediate claim Hrlt: Rlt (mul_SNo a c) t.
An exact proof term for the current goal is (order_rel_R_implies_Rlt (mul_SNo a c) t Hrel).
We prove the intermediate claim HtS: SNo t.
An exact proof term for the current goal is (real_SNo t HtR).
We prove the intermediate claim HcS: SNo c.
An exact proof term for the current goal is (real_SNo c HcR).
We prove the intermediate claim HaS: SNo a.
An exact proof term for the current goal is (real_SNo a HaR).
We prove the intermediate claim HacR: mul_SNo a c ∈ R.
An exact proof term for the current goal is (real_mul_SNo a HaR c HcR).
We prove the intermediate claim HacS: SNo (mul_SNo a c).
An exact proof term for the current goal is (real_SNo (mul_SNo a c) HacR).
We prove the intermediate claim HmulLt: mul_SNo a c < t.
An exact proof term for the current goal is (RltE_lt (mul_SNo a c) t Hrlt).
We prove the intermediate claim Halt: a < div_SNo t c.
An exact proof term for the current goal is (div_SNo_pos_LtR t c a HtS HcS HaS Hcpos HmulLt).
We prove the intermediate claim Hray: order_rel R a (div_SNo t c).
An exact proof term for the current goal is (Rlt_implies_order_rel_R a (div_SNo t c) (RltI a (div_SNo t c) HaR (real_div_SNo t HtR c HcR) Halt)).
We prove the intermediate claim Himg: div_SNo t c ∈ open_ray_upper R a.
An exact proof term for the current goal is (SepI R (λx0 : set ⇒ order_rel R a x0) (div_SNo t c) (real_div_SNo t HtR c HcR) Hray).
We prove the intermediate claim Hpredef: preimage_of R (div_const_fun c) (open_ray_upper R a) = {x0 ∈ R|apply_fun (div_const_fun c) x0 ∈ open_ray_upper R a}.
Use reflexivity.
rewrite the current goal using Hpredef (from left to right).
Apply (SepI R (λx0 : set ⇒ apply_fun (div_const_fun c) x0 ∈ open_ray_upper R a) t HtR) to the current goal.
rewrite the current goal using (div_const_fun_apply c t HcR HtR) (from left to right).
An exact proof term for the current goal is Himg.
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