Let c and b be given.
Assume HcR: c ∈ R.
Assume Hcpos: 0 < c.
Assume HbR: b ∈ R.
We will prove preimage_of R (mul_const_fun c) (open_ray_lower R b) = open_ray_lower R (div_SNo b c).
Apply set_ext to the current goal.
Let t be given.
Assume HtPre: t ∈ preimage_of R (mul_const_fun c) (open_ray_lower R b).
We will prove t ∈ open_ray_lower R (div_SNo b c).
We prove the intermediate claim HtR: t ∈ R.
An exact proof term for the current goal is (SepE1 R (λx0 : set ⇒ apply_fun (mul_const_fun c) x0 ∈ open_ray_lower R b) t HtPre).
We prove the intermediate claim Himg: apply_fun (mul_const_fun c) t ∈ open_ray_lower R b.
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ apply_fun (mul_const_fun c) x0 ∈ open_ray_lower R b) t HtPre).
We prove the intermediate claim Hrel: order_rel R (apply_fun (mul_const_fun c) t) b.
An exact proof term for the current goal is (SepE2 R (λy0 : set ⇒ order_rel R y0 b) (apply_fun (mul_const_fun c) t) Himg).
We prove the intermediate claim Hlt: Rlt (apply_fun (mul_const_fun c) t) b.
An exact proof term for the current goal is (order_rel_R_implies_Rlt (apply_fun (mul_const_fun c) t) b 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 HbS: SNo b.
An exact proof term for the current goal is (real_SNo b HbR).
We prove the intermediate claim HmulEq: apply_fun (mul_const_fun c) t = mul_SNo t c.
rewrite the current goal using (mul_const_fun_apply c t HcR HtR) (from left to right).
Use reflexivity.
We prove the intermediate claim HltS: mul_SNo t c < b.
rewrite the current goal using HmulEq (from right to left) at position 1.
An exact proof term for the current goal is (RltE_lt (apply_fun (mul_const_fun c) t) b Hlt).
We prove the intermediate claim HtLtS: t < div_SNo b c.
An exact proof term for the current goal is (div_SNo_pos_LtR b c t HbS HcS HtS Hcpos HltS).
We prove the intermediate claim HdivR: div_SNo b c ∈ R.
An exact proof term for the current goal is (real_div_SNo b HbR c HcR).
We prove the intermediate claim HtLt: Rlt t (div_SNo b c).
An exact proof term for the current goal is (RltI t (div_SNo b c) HtR HdivR HtLtS).
We prove the intermediate claim Hray: order_rel R t (div_SNo b c).
An exact proof term for the current goal is (Rlt_implies_order_rel_R t (div_SNo b c) HtLt).
An exact proof term for the current goal is (SepI R (λx0 : set ⇒ order_rel R x0 (div_SNo b c)) t HtR Hray).
Let t be given.
Assume HtRay: t ∈ open_ray_lower R (div_SNo b c).
We will prove t ∈ preimage_of R (mul_const_fun c) (open_ray_lower R b).
We prove the intermediate claim HtR: t ∈ R.
An exact proof term for the current goal is (SepE1 R (λx0 : set ⇒ order_rel R x0 (div_SNo b c)) t HtRay).
We prove the intermediate claim Hrel: order_rel R t (div_SNo b c).
An exact proof term for the current goal is (SepE2 R (λx0 : set ⇒ order_rel R x0 (div_SNo b c)) t HtRay).
We prove the intermediate claim HtLt: Rlt t (div_SNo b c).
An exact proof term for the current goal is (order_rel_R_implies_Rlt t (div_SNo b c) 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 HbS: SNo b.
An exact proof term for the current goal is (real_SNo b HbR).
We prove the intermediate claim HtLtS: t < div_SNo b c.
An exact proof term for the current goal is (RltE_lt t (div_SNo b c) HtLt).
We prove the intermediate claim HmulLtS: mul_SNo t c < b.
An exact proof term for the current goal is (div_SNo_pos_LtR' b c t HbS HcS HtS Hcpos HtLtS).
We prove the intermediate claim HmulR: mul_SNo t c ∈ R.
An exact proof term for the current goal is (real_mul_SNo t HtR c HcR).
We prove the intermediate claim HmulLt: Rlt (mul_SNo t c) b.
An exact proof term for the current goal is (RltI (mul_SNo t c) b HmulR HbR HmulLtS).
We prove the intermediate claim Hray: order_rel R (mul_SNo t c) b.
An exact proof term for the current goal is (Rlt_implies_order_rel_R (mul_SNo t c) b HmulLt).
We prove the intermediate claim Himg: mul_SNo t c ∈ open_ray_lower R b.
An exact proof term for the current goal is (SepI R (λy0 : set ⇒ order_rel R y0 b) (mul_SNo t c) HmulR Hray).
We prove the intermediate claim Hpredef: preimage_of R (mul_const_fun c) (open_ray_lower R b) = {x0 ∈ R|apply_fun (mul_const_fun c) x0 ∈ open_ray_lower R b}.
Use reflexivity.
rewrite the current goal using Hpredef (from left to right).
Apply (SepI R (λx0 : set ⇒ apply_fun (mul_const_fun c) x0 ∈ open_ray_lower R b) t HtR) to the current goal.
rewrite the current goal using (mul_const_fun_apply c t HcR HtR) (from left to right).
An exact proof term for the current goal is Himg.
∎