Mathlib.Geometry.Affine.Derived

import Mathlib.Algebra.Ring.Basic
import Mathlib.Algebra.OrderedField.Basic
import Mathlib.Algebra.OrderedField.Square
import Mathlib.LinearAlgebra.VectorSpace
import Mathlib.LinearAlgebra.InnerProduct
import Mathlib.Geometry.Affine

theorem vec_add_comm_from_vector_args.{u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun vector_args => fun x => fun y => vector_args (@Eq.{v} Vector (vadd x y) (vadd y x)) (fun (vec_sub_def_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))) => fun (vec_add_assoc_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (vadd x y) z) (vadd x (vadd y z))) => fun (vec_add_comm_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (vadd x y) (vadd y x)) => fun (vec_add_zero_arg : forall (x : Vector), @Eq.{v} Vector (vadd x vzero) x) => fun (vec_zero_add_arg : forall (x : Vector), @Eq.{v} Vector (vadd vzero x) x) => fun (vec_neg_add_cancel_arg : forall (x : Vector), @Eq.{v} Vector (vadd (vneg x) x) vzero) => fun (vec_add_neg_cancel_arg : forall (x : Vector), @Eq.{v} Vector (vadd x (vneg x)) vzero) => fun (sub_sub_sub_cancel_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg y z)) (@vsub.{v} Vector vadd vneg x y)) => fun (vec_sub_self_arg : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x x) vzero) => fun (vec_sub_zero_arg : forall (x : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x vzero) x) => fun (vec_add_left_cancel_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), forall (h : @Eq.{v} Vector (vadd x y) (vadd x z)), @Eq.{v} Vector y z) => fun (smul_add_arg : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (add a b) x) (vadd (smul a x) (smul b x))) => fun (add_smul_arg : forall (a : Scalar), forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (smul a (vadd x y)) (vadd (smul a x) (smul a y))) => fun (one_smul_arg : forall (x : Vector), @Eq.{v} Vector (smul one x) x) => fun (mul_smul_arg : forall (a : Scalar), forall (b : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (mul a b) x) (smul a (smul b x))) => fun (zero_smul_arg : forall (x : Vector), @Eq.{v} Vector (smul zero x) vzero) => fun (smul_zero_arg : forall (a : Scalar), @Eq.{v} Vector (smul a vzero) vzero) => fun (neg_smul_arg : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul (neg a) x) (vneg (smul a x))) => fun (smul_neg_arg : forall (a : Scalar), forall (x : Vector), @Eq.{v} Vector (smul a (vneg x)) (vneg (smul a x))) => fun (vec_sub_eq_add_neg_arg : forall (x : Vector), forall (y : Vector), @Eq.{v} Vector (@vsub.{v} Vector vadd vneg x y) (vadd x (vneg y))) => fun (sub_add_sub_cancel_left_arg : forall (x : Vector), forall (y : Vector), forall (z : Vector), @Eq.{v} Vector (vadd (@vsub.{v} Vector vadd vneg x z) (@vsub.{v} Vector vadd vneg z y)) (@vsub.{v} Vector vadd vneg x y)) => fun (linear_comb2_ext_arg : forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), @Eq.{v} Vector (@linear_comb2.{u,v} Scalar Vector vadd smul a x b y) (vadd (smul a x) (smul b y))) => fun (linear_comb3_ext_arg : forall (a : Scalar), forall (x : Vector), forall (b : Scalar), forall (y : Vector), forall (c : Scalar), forall (z : Vector), @Eq.{v} Vector (@linear_comb3.{u,v} Scalar Vector vadd smul a x b y c z) (vadd (vadd (smul a x) (smul b y)) (smul c z))) => vec_add_comm_arg x y)

theorem disp_reverse_from_affine_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => affine_args (@Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (fun (disp_self_arg : forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero) => fun (disp_reverse_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) => fun (disp_comp_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))) => fun (point_ext_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B) => fun (dist_sq_symm_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)) => fun (dist_sq_zero_iff_eq_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R) => disp_reverse_arg A B)

theorem disp_comp_from_affine_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => fun C => affine_args (@Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))) (fun (disp_self_arg : forall (A : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A A) vzero) => fun (disp_reverse_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) => fun (disp_comp_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A C) (vadd (@disp.{p,v} PointCarrier Vector disp_op A B) (@disp.{p,v} PointCarrier Vector disp_op B C))) => fun (point_ext_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (h : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op A B) vzero), @Eq.{p} PointCarrier A B) => fun (dist_sq_symm_arg : forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B A)) => fun (dist_sq_zero_iff_eq_arg : forall (A : PointCarrier), forall (B : PointCarrier), forall (R : Prop), forall (mk : forall (forward : forall (h : @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), @Eq.{p} PointCarrier A B), forall (backward : forall (h : @Eq.{p} PointCarrier A B), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) zero), R), R) => disp_comp_arg A B C)

theorem dist_sq_points_def_from_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => @Eq.refl.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B)

theorem hypotenuse_vector_eq_neg_left_add_right_from_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => fun C => @Eq.rec.{v,0} Vector (vadd (@disp.{p,v} PointCarrier Vector disp_op B A) (@disp.{p,v} PointCarrier Vector disp_op A C)) (fun (z : Vector) => fun (hz : @Eq.{v} Vector (vadd (@disp.{p,v} PointCarrier Vector disp_op B A) (@disp.{p,v} PointCarrier Vector disp_op A C)) z) => @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) z) (@disp_comp_from_affine_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B A C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) (@Eq.rec.{v,0} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) (fun (q : Vector) => fun (hq : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B A) q) => @Eq.{v} Vector (vadd (@disp.{p,v} PointCarrier Vector disp_op B A) (@disp.{p,v} PointCarrier Vector disp_op A C)) (vadd q (@disp.{p,v} PointCarrier Vector disp_op A C))) (@Eq.refl.{v} Vector (vadd (@disp.{p,v} PointCarrier Vector disp_op B A) (@disp.{p,v} PointCarrier Vector disp_op A C))) (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp_reverse_from_affine_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B))

theorem hypotenuse_vector_eq_sub_legs_from_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B)) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun vector_args => fun affine_args => fun A => fun B => fun C => @Eq.rec.{v,0} Vector (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) (fun (z : Vector) => fun (hz : @Eq.{v} Vector (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) z) => @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) z) (@hypotenuse_vector_eq_neg_left_add_right_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B C) (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B)) (@vec_add_comm_from_vector_args.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul vector_args (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))

theorem dist_sq_hypotenuse_norm_neg_left_add_right_from_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun affine_args => fun A => fun B => fun C => @Eq.rec.{v,0} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (fun (z : Vector) => fun (hz : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) z) => @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner z)) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B C) (vadd (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) (@hypotenuse_vector_eq_neg_left_add_right_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args A B C)

theorem dist_sq_hypotenuse_norm_sub_legs_from_args.{p,u,v} :
  forall (Scalar : Sort u), forall (zero : Scalar), forall (one : Scalar), forall (add : forall (a : Scalar), forall (b : Scalar), Scalar), forall (neg : forall (a : Scalar), Scalar), forall (sub : forall (a : Scalar), forall (b : Scalar), Scalar), forall (mul : forall (a : Scalar), forall (b : Scalar), Scalar), forall (le_rel : forall (a : Scalar), forall (b : Scalar), Prop), forall (Vector : Sort v), forall (vzero : Vector), forall (vadd : forall (x : Vector), forall (y : Vector), Vector), forall (vneg : forall (x : Vector), Vector), forall (smul : forall (a : Scalar), forall (x : Vector), Vector), forall (inner : forall (x : Vector), forall (y : Vector), Scalar), forall (PointCarrier : Sort p), forall (disp_op : forall (A : PointCarrier), forall (B : PointCarrier), Vector), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (affine_args : @AffineLawArgs.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op), forall (A : PointCarrier), forall (B : PointCarrier), forall (C : PointCarrier), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B))) :=
  fun Scalar => fun zero => fun one => fun add => fun neg => fun sub => fun mul => fun le_rel => fun Vector => fun vzero => fun vadd => fun vneg => fun smul => fun inner => fun PointCarrier => fun disp_op => fun vector_args => fun affine_args => fun A => fun B => fun C => @Eq.rec.{v,0} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) (fun (z : Vector) => fun (hz : @Eq.{v} Vector (@disp.{p,v} PointCarrier Vector disp_op B C) z) => @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner z)) (@dist_sq_points_def_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op affine_args B C) (@vsub.{v} Vector vadd vneg (@disp.{p,v} PointCarrier Vector disp_op A C) (@disp.{p,v} PointCarrier Vector disp_op A B)) (@hypotenuse_vector_eq_sub_legs_from_args.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op vector_args affine_args A B C)