diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Ray_ex.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Ray_ex.lean
index e81ed044..7e78439f 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Ray_ex.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Ray_ex.lean
@@ -18,12 +18,19 @@ def Seg.reverse (seg : Seg P): Seg P where
   target := seg.source
 
 -- reversing the toDir does not change the nontriviality of a generalized directed segment.
-theorem nd_of_rev_of_nd {seg : Seg P} (nd : seg.is_nd) : seg.reverse.is_nd := sorry
+theorem nd_of_rev_of_nd {seg : Seg P} (nd : seg.is_nd) : seg.reverse.is_nd := by
+-- sorry
+  simp only [Seg.is_nd]
+  push_neg
+  symm
+  apply nd
 
 def Seg_nd.reverse : Seg_nd P := ⟨seg_nd.1.reverse, nd_of_rev_of_nd seg_nd.2⟩ 
 
 -- double reverse a ray gets back to itself
-theorem Ray.rev_rev_eq_self : ray.reverse.reverse = ray := sorry
+theorem Ray.rev_rev_eq_self : ray.reverse.reverse = ray := by
+-- sorry
+  simp only [reverse, neg_neg]
 
 -- double reverse a generalized directed segment gets back to itself
 theorem Seg.rev_rev_eq_self  : seg.reverse.reverse = seg := rfl
@@ -33,35 +40,227 @@ theorem Seg_nd.rev_rev_eq_self  : seg_nd.reverse.reverse = seg_nd := rfl
 theorem Seg_nd.rev_toSeg_eq_toSeg_rev : seg_nd.1.reverse = seg_nd.reverse.1 := rfl
 
 -- reversing the toDir does not change the property that a point lies on the generalized directed segments.
-theorem Seg.lies_on_rev_iff_lie_son {A : P} (lieson : A LiesOn seg) : A LiesOn seg.reverse := sorry
-
-theorem eq_source_iff_lies_on_ray_lies_on_ray_rev {A : P} : A = ray.source ↔ (A LiesOn ray) ∧ (A LiesOn ray.reverse) := sorry
-
-theorem not_lies_on_of_lies_int_rev {A : P} (liesint : A LiesInt ray.reverse) : ¬ A LiesOn ray := sorry
-
-theorem not_lies_int_of_lies_on_rev {A : P} (liesint : A LiesOn ray.reverse) : ¬ A LiesInt ray := sorry
-
--- A point lies on the directed segment if and only if it lies on the ray associated to the segment and the ray associated to the reverse of this segment.
-theorem lies_on_iff_lies_on_toRay_and_rev_toRay {A : P} : A LiesOn seg_nd.1 ↔ (A LiesOn seg_nd.toRay) ∧ (A LiesOn seg_nd.reverse.toRay) := sorry
+theorem Seg.lies_on_rev_iff_lie_son {A : P} (lieson : A LiesOn seg) : A LiesOn seg.reverse := by
+-- sorry
+  unfold lies_on Carrier.carrier instCarrierSeg
+  simp only [Set.setOf_mem_eq]
+  rcases lieson with ⟨a,⟨h1,⟨h2,h3⟩⟩⟩
+  use 1-a
+  constructor
+  linarith
+  constructor
+  linarith
+  simp only [reverse]
+  rw[sub_smul,sub_eq_add_neg]
+  simp only [one_smul]
+  have h:-(a • VEC seg.target seg.source)=a • (-1) • VEC seg.target seg.source:=by
+    rw[smul_algebra_smul_comm]
+    rw[neg_smul]
+    simp only [Complex.real_smul, one_smul]
+  have h':VEC seg.target A=VEC seg.target seg.source+VEC seg.source A:=by
+    rw[←neg_vec seg.source seg.target,add_comm,←sub_eq_add_neg,vec_sub_vec]
+  rw[h]
+  rw[h',←neg_vec seg.source seg.target]
+  simp only [smul_neg, neg_smul, one_smul, neg_neg, Complex.real_smul, add_right_inj]
+  exact h3
+  
+theorem eq_source_iff_lies_on_ray_lies_on_ray_rev {A : P} : A = ray.source ↔ (A LiesOn ray) ∧ (A LiesOn ray.reverse) := by
+-- sorry
+  constructor
+  intro h
+  constructor
+  use 0
+  simp only [le_refl, zero_smul, true_and]
+  rw[h,vec_same_eq_zero]
+  use 0
+  simp only [le_refl, Dir.toVec_neg_eq_neg_toVec, smul_neg, zero_smul, neg_zero, true_and]
+  simp only [Ray.reverse]
+  rw[h,vec_same_eq_zero]
+  simp only [and_imp]
+  rintro ⟨a,⟨anneg,h⟩⟩ ⟨b,⟨bnneg,h'⟩⟩
+  simp only [Ray.reverse] at h'
+  simp only [Dir.toVec_neg_eq_neg_toVec, smul_neg] at h' 
+  rw[h,←add_zero a,← sub_self b,add_sub,sub_smul] at h'
+  simp only [sub_eq_neg_self, mul_eq_zero] at h' 
+  have h'': a+b=0:=by
+    contrapose! h'
+    apply smul_ne_zero
+    exact h'
+    apply Dir.toVec_ne_zero
+  have:a=0:=by
+    linarith
+  rw[this] at h
+  simp only [zero_smul] at h 
+  rw[eq_iff_vec_eq_zero,h]
+
+theorem not_lies_on_of_lies_int_rev {A : P} (liesint : A LiesInt ray.reverse) : ¬ A LiesOn ray := by
+-- sorry
+  by_contra h
+  rcases liesint with ⟨h',nsource⟩
+  have: A LiesOn ray.reverse:=by
+    apply h'
+  have :A=ray.source:=by
+    rw[eq_source_iff_lies_on_ray_lies_on_ray_rev]
+    constructor
+    exact h
+    exact this
+  trivial
+
+theorem not_lies_int_of_lies_on_rev {A : P} (liesint : A LiesOn ray.reverse) : ¬ A LiesInt ray :=by
+--  sorry
+  by_contra h
+  rw[←Ray.rev_rev_eq_self ray] at h
+  have:¬A LiesOn ray.reverse:=by
+    apply not_lies_on_of_lies_int_rev 
+    exact h
+  trivial
 
 -- `This theorem really concerns about the total order on a line`
 theorem lies_on_pt_toDir_of_pt_lies_on_rev {A B : P} {ray : Ray P} (hA : A LiesOn ray) (hB : B LiesOn ray.reverse) : A LiesOn Ray.mk B ray.toDir := sorry
 
 theorem Ray.toDir_of_reverse_eq_neg_toDir : ray.reverse.toDir = - ray.toDir := rfl
 
-theorem Ray.toProj_of_reverse_eq_toProj : ray.reverse.toProj = ray.toProj := sorry
-
-theorem Seg.toVec_of_reverse_eq_neg_toVec : seg.reverse.toVec = - seg.toVec := sorry
-
-theorem Seg_nd.toVec_nd_of_reverse_eq_neg_toVec_nd : seg_nd.reverse.toVec_nd = - seg_nd.toVec_nd := sorry
-
-theorem Seg_nd.toDir_of_reverse_eq_neg_toDir : seg_nd.reverse.toDir = - seg_nd.toDir := sorry
-
-theorem Seg_nd.toProj_of_reverse_eq_toProj : seg_nd.reverse.toProj = seg_nd.toProj := sorry
+theorem Ray.toProj_of_reverse_eq_toProj : ray.reverse.toProj = ray.toProj := by
+-- sorry
+  --@HeRunming: Simply imitate the proof of theorem "eq_toProj_of_smul" in Vector.lean
+  unfold Ray.reverse
+  unfold Ray.toDir
+  unfold Ray.toProj
+  apply Quotient.sound
+  unfold HasEquiv.Equiv instHasEquiv PM.con PM
+  simp only [Con.rel_eq_coe, Con.rel_mk]
+  unfold EuclidGeom.Ray.toDir
+  right
+  rfl
+  
+theorem Seg.toVec_of_reverse_eq_neg_toVec : seg.reverse.toVec = - seg.toVec := by
+-- sorry
+  unfold toVec reverse
+  simp only [reverse]
+  rw[neg_vec]
+
+theorem Seg_nd.toVec_nd_of_reverse_eq_neg_toVec_nd : seg_nd.reverse.toVec_nd = - seg_nd.toVec_nd := by
+-- sorry
+  apply Subtype.eq
+  apply Seg.toVec_of_reverse_eq_neg_toVec
+  -- simp
+
+theorem Seg_nd.toDir_of_reverse_eq_neg_toDir : seg_nd.reverse.toDir = - seg_nd.toDir := by
+-- sorry
+  unfold toDir
+  symm
+  have :seg_nd.reverse.toVec_nd.1=(-1)•seg_nd.toVec_nd.1:=by
+    rw[neg_smul,one_smul]
+    rw[Seg_nd.toVec_nd_of_reverse_eq_neg_toVec_nd]
+    rfl
+  apply @neg_normalize_eq_normalize_smul_neg _ _ (-1)
+  rw[this]
+  simp only [ne_eq, neg_smul, one_smul]
+  norm_num
+
+theorem Seg_nd.toProj_of_reverse_eq_toProj : seg_nd.reverse.toProj = seg_nd.toProj := by
+-- sorry
+  apply @eq_toProj_of_smul _ _ (-1)
+  rw[neg_smul,one_smul]
+  rw[Seg_nd.toVec_nd_of_reverse_eq_neg_toVec_nd]
+  apply neg_eq_iff_eq_neg.mp
+  rfl
 
 -- reversing the toDir does not change the length
-theorem length_eq_length_of_rev : seg.length = seg.reverse.length := sorry
+theorem length_eq_length_of_rev : seg.length = seg.reverse.length := by
+-- sorry
+  unfold Seg.length
+  have h:Vec.norm seg.toVec=norm seg.toVec:=by
+    rfl
+  have h':Vec.norm seg.reverse.toVec=norm seg.reverse.toVec:=by
+    rfl
+  rw[←h,←h']
+  rw[Seg.toVec_of_reverse_eq_neg_toVec]
+  have eq: (-Seg.toVec seg)=-(Seg.toVec seg):=by
+    rfl
+  rw[eq]
+  simp only [Vec.norm]
+  norm_num
 
+-- A point lies on the directed segment if and only if it lies on the ray associated to the segment and the ray associated to the reverse of this segment.
+-- need to be refined
+theorem lies_on_iff_lies_on_toRay_and_rev_toRay {A : P} : A LiesOn seg_nd.1 ↔ (A LiesOn seg_nd.toRay) ∧ (A LiesOn seg_nd.reverse.toRay) := by
+-- sorry
+  constructor
+  intro liesonseg
+  constructor
+  apply Seg_nd.lies_on_toRay_of_lies_on
+  trivial
+  apply Seg_nd.lies_on_toRay_of_lies_on
+  apply Seg.lies_on_rev_iff_lie_son
+  trivial
+  rintro ⟨⟨a,anneg,h⟩,b,bnneg,h'⟩
+  unfold Dir.toVec Ray.toDir Seg_nd.toRay at h h'
+  rw[Seg_nd.toDir_of_reverse_eq_neg_toDir] at h'
+  have tria:(-Seg_nd.toDir seg_nd).1=(-1)•(Seg_nd.toDir seg_nd).1:=by
+    rw[Dir.toVec_neg_eq_neg_toVec _]
+    rw[neg_one_smul]
+  unfold Dir.toVec at tria
+  rw[tria] at h'
+  simp only [Seg_nd.toDir] at h h'
+  have trib:b • -1 • (Vec_nd.normalize (Seg_nd.toVec_nd seg_nd)).1=(-b)• (Vec_nd.normalize (Seg_nd.toVec_nd seg_nd)).1:=by
+    simp only [neg_smul, one_smul, smul_neg, Complex.real_smul]
+  unfold Dir.toVec at trib
+  rw[trib] at h'
+  set v:=(Vec_nd.normalize (Seg_nd.toVec_nd seg_nd)).1 with v_def
+  unfold Dir.toVec at v_def
+  simp only [Seg_nd.reverse,Seg.reverse] at h'
+  rw[←v_def] at h h'
+  have asumbv:(a+b)•v=VEC seg_nd.1.source seg_nd.1.target:=by
+    rw[← vec_add_vec _ A _,←neg_vec seg_nd.1.target A,h,h',add_smul]
+    simp
+  have tri1:VEC seg_nd.1.source seg_nd.1.target=seg_nd.toVec_nd.1:=by
+    rfl
+  rw[tri1,←Vec_nd.norm_smul_normalize_eq_self seg_nd.toVec_nd] at asumbv
+  have tri2:(a+b-(Vec.norm seg_nd.toVec_nd))•(seg_nd.toDir).1=0:=by
+    rw[sub_smul,asumbv]
+    simp
+  have asumb:a+b=(Vec.norm seg_nd.toVec_nd):=by
+    have asumb:a+b-(Vec.norm seg_nd.toVec_nd)=0:=by
+      rcases smul_eq_zero.mp tri2 with hyp1|hyp2
+      exact hyp1
+      exfalso
+      apply Dir.toVec_ne_zero seg_nd.toDir
+      assumption
+    linarith
+  have norm_pos_vec:0<Vec.norm seg_nd.toVec_nd:=by
+    simp only [ne_eq]
+    apply norm_pos_iff.mpr (seg_nd.toVec_nd.2)
+  have norm_nonzero:Vec.norm seg_nd.toVec_nd≠0:=by
+    linarith
+  have alenorm:a≤Vec.norm seg_nd.toVec_nd:=by
+    linarith
+  have tri3:1=Vec.norm seg_nd.toVec_nd*(Vec.norm seg_nd.toVec_nd)⁻¹:=by
+    rw[mul_inv_cancel norm_nonzero]
+  use a*(Vec.norm seg_nd.toVec_nd)⁻¹
+  constructor
+  apply mul_nonneg
+  exact anneg
+  simp only [ne_eq, inv_nonneg]
+  linarith
+  constructor
+  rw[tri3]
+  apply mul_le_mul
+  exact alenorm
+  trivial
+  simp only [ne_eq, inv_nonneg]
+  apply le_trans anneg
+  exact alenorm
+  apply le_trans anneg
+  exact alenorm
+  rw[h,mul_smul,tri1]
+  nth_rw 3[←Vec_nd.norm_smul_normalize_eq_self]
+  simp only [v_def]
+  rw[smul_smul,smul_smul,mul_assoc,←smul_smul]
+  rw[mul_comm] at tri3
+  rw[←tri3]
+  simp only [Complex.real_smul, one_smul]
+  
 end reverse
 
 section extension
@@ -69,12 +268,100 @@ section extension
 -- Define the extension ray from a nontrival segment
 def Seg_nd.extension : Ray P := (seg_nd.reverse.toRay).reverse
 
-theorem extn_eq_rev_toray_rev : seg_nd.extension = seg_nd.reverse.toRay.reverse := sorry
-
-theorem eq_target_iff_lies_on_lies_on_extn {A : P} : (A LiesOn seg_nd.1) ∧ (A LiesOn seg_nd.extension) ↔ A = seg_nd.1.target := sorry
-
-theorem target_lies_int_seg_source_pt_of_pt_lies_int_extn {A : P} (liesint : A LiesInt seg_nd.extension) : seg_nd.1.target LiesInt SEG seg_nd.1.source A := sorry
-
+theorem extn_eq_rev_toray_rev : seg_nd.extension = seg_nd.reverse.toRay.reverse :=by
+-- sorry
+  --@HeRunming
+  rfl
+
+theorem eq_target_iff_lies_on_lies_on_extn {A : P} : (A LiesOn seg_nd.1) ∧ (A LiesOn seg_nd.extension) ↔ A = seg_nd.1.target := by
+-- sorry
+  constructor
+  rintro ⟨hyp1,hyp2⟩
+  have h:seg_nd.1.target=seg_nd.extension.source :=by rfl
+  rw[h]
+  apply (eq_source_iff_lies_on_ray_lies_on_ray_rev seg_nd.extension).mpr
+  constructor
+  exact hyp2
+  have h':seg_nd.reverse.toRay=seg_nd.extension.reverse:=by
+    unfold Ray.reverse Seg_nd.toRay Ray.source Seg.source Seg_nd.reverse Seg.reverse
+    simp only [Seg_nd.extension,Ray.reverse]
+    unfold Ray.source Seg_nd.reverse Seg.reverse Ray.toDir Seg_nd.toRay
+    simp only [neg_neg]
+  rw[←h']
+  apply Seg_nd.lies_on_toRay_of_lies_on
+  apply Seg.lies_on_rev_iff_lie_son
+  apply hyp1
+  intro hyp
+  constructor
+  use 1
+  simp only [zero_le_one, le_refl, one_smul, true_and]
+  rw[hyp]
+  use 0
+  simp only [le_refl, Dir.toVec_neg_eq_neg_toVec, smul_neg, zero_smul, neg_zero, true_and]
+  rw[hyp]
+  have :(Seg_nd.extension seg_nd).source=seg_nd.1.target:=by
+    rfl
+  rw[this]
+  simp only [vec_same_eq_zero]
+
+theorem target_lies_int_seg_source_pt_of_pt_lies_int_extn {A : P} (liesint : A LiesInt seg_nd.extension) : seg_nd.1.target LiesInt SEG seg_nd.1.source A :=
+by
+-- sorry
+  rcases liesint with ⟨⟨a,anonneg,ha⟩,nonsource⟩
+  have raysourcesegtarget:seg_nd.1.target=seg_nd.extension.1:=by
+    rfl
+  have sourcetargetA:VEC seg_nd.1.source seg_nd.1.target+VEC seg_nd.1.target A=VEC seg_nd.1.source A:=by
+    rw[vec_add_vec]
+  have vec_ndtovec:VEC seg_nd.1.source seg_nd.1.target=seg_nd.toVec_nd.1:=by
+    rfl
+  have apos:0<a:=by
+    contrapose! nonsource
+    have:a=0:=by linarith
+    rw[this] at ha
+    simp only [Dir.toVec_neg_eq_neg_toVec, smul_neg, zero_smul, neg_zero] at ha
+    apply (eq_iff_vec_eq_zero _ _).mpr
+    exact ha
+  have raysourcesource:seg_nd.extension.source=seg_nd.1.target:=by
+    rfl
+  have seg_pos:0< Vec_nd.norm (Seg_nd.toVec_nd seg_nd):=by
+    simp only [ne_eq, norm_of_Vec_nd_eq_norm_of_Vec_nd_fst,Vec.norm]
+    apply norm_pos_iff.mpr (seg_nd.toVec_nd.2)
+  have seg_nonzero:Vec_nd.norm (Seg_nd.toVec_nd seg_nd)≠0:=by linarith
+  have aseg_pos:0<Vec_nd.norm (Seg_nd.toVec_nd seg_nd)+a:=by
+    linarith
+  have aseg_nonzero:Vec_nd.norm (Seg_nd.toVec_nd seg_nd)+a≠ 0:=by
+    linarith
+  have raydir:seg_nd.extension.toDir.toVec=seg_nd.toVec_nd.normalize.toVec:=by
+    rw[Ray.toDir_of_reverse_eq_neg_toDir,←Seg_nd.toDir_eq_toRay_toDir,Seg_nd.toDir_of_reverse_eq_neg_toDir]
+    simp only [neg_neg]
+  have segleaseg:Vec_nd.norm (Seg_nd.toVec_nd seg_nd)≤Vec_nd.norm (Seg_nd.toVec_nd seg_nd)+a:=by
+    linarith
+  constructor
+  use (seg_nd.toVec_nd.norm)*(seg_nd.toVec_nd.norm+a)⁻¹
+  constructor
+  apply mul_nonneg
+  linarith[seg_pos]
+  norm_num
+  rw[←norm_of_Vec_nd_eq_norm_of_Vec_nd_fst]
+  linarith
+  constructor
+  rw[←mul_inv_cancel aseg_nonzero]
+  apply mul_le_mul
+  linarith
+  linarith
+  norm_num
+  rw[← norm_of_Vec_nd_eq_norm_of_Vec_nd_fst]
+  linarith
+  linarith
+  simp only [Seg.target]
+  rw[←raysourcesegtarget] at ha
+  rw[←sourcetargetA,ha,vec_ndtovec,←Vec_nd.norm_smul_normalize_eq_self (seg_nd.toVec_nd),←norm_of_Vec_nd_eq_norm_of_Vec_nd_fst,raydir]
+  rw[←add_smul,← mul_smul,mul_assoc,inv_mul_cancel,mul_one]
+  linarith
+  constructor
+  exact seg_nd.2
+  rw[←raysourcesegtarget] at nonsource
+  symm
+  exact nonsource
 end extension
-
 end EuclidGeom
\ No newline at end of file