pythagorean_distance_sq_from_law_packages

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 (ring_args : @RingLawArgs.{u} Scalar zero one add neg sub mul), forall (vector_args : @VectorSpaceLawArgs.{u,v} Scalar zero one add neg sub mul Vector vzero vadd vneg smul), forall (inner_args : @InnerProductLawArgs.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner), 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), forall (h : @RightTriangle.{p,u,v} Scalar zero Vector inner PointCarrier disp_op A B C), @Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier 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 ring_args => fun vector_args => fun inner_args => fun affine_args => fun A => fun B => fun C => fun h => @right_triangle_affine_additive_perp_bridge_from_rt.{p,u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner PointCarrier disp_op ring_args inner_args affine_args A B C h (@Eq.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))) (fun (hypotenuse_orientation : @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 (perp_premise : @PerpVec.{u,v} Scalar zero Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C)) => @eq_calc3.{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))) (add (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector disp_op A C))) (add (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A B) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C)) (@eq_trans.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op B C) (@normSq.{u,v} Scalar Vector inner (@disp.{p,v} PointCarrier Vector 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))) (@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) (@eq_congr_arg.{v,u} Vector Scalar (fun (x : Vector) => @normSq.{u,v} Scalar Vector inner x) (@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)) hypotenuse_orientation)) (@norm_sq_add_of_perp_from_args.{u,v} Scalar zero one add neg sub mul le_rel Vector vzero vadd vneg smul inner ring_args inner_args (vneg (@disp.{p,v} PointCarrier Vector disp_op A B)) (@disp.{p,v} PointCarrier Vector disp_op A C) perp_premise) (@eq_congr2.{u,u,u} Scalar Scalar Scalar add (@normSq.{u,v} Scalar Vector inner (vneg (@disp.{p,v} PointCarrier Vector disp_op A B))) (@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 C)) (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C) (@pythagorean_left_leg_norm_neg_from_law_packages.{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) (@Eq.refl.{u} Scalar (@distSqPoints.{p,u,v} Scalar Vector inner PointCarrier disp_op A C))))