for b1 being _Graph for b2 being Walk of b1 for b3, b4 being Nat holds ( ( not b3 is even & not b4 is even & b3<= b4 & b4<=len b2 implies b2.cut b3,b4= b3,b4-cut b2 ) & ( not ( not b3 is even & not b4 is even & b3<= b4 & b4<=len b2 ) implies b2.cut b3,b4= b2 ) );
for b1 being _Graph for b2 being Walk of b1 for b3 being EdgeSeq of b1 holds ( b3= b2.edgeSeq() iff ( len b2=(2 *(len b3))+ 1 & ( for b4 being Nat holds ( 1 <= b4 & b4<=len b3 implies b3. b4= b2.(2 * b4) ) ) ) );
for b1 being _Graph for b2 being Walk of b1 for b3 being Nat for b4 being oddNat holds ( ( not b3 is even & b3<=len b2 implies ( b4= b2.find b3 iff ( b4<=len b2 & b2. b4= b2. b3 & ( for b5 being oddNat holds ( b5<=len b2 & b2. b5= b2. b3 implies b4<= b5 ) ) ) ) ) & ( not ( not b3 is even & b3<=len b2 ) implies ( b4= b2.find b3 iff b4=len b2 ) ) );
for b1 being _Graph for b2 being Walk of b1 for b3 being Nat for b4 being oddNat holds ( ( not b3 is even & b3<=len b2 implies ( b4= b2.rfind b3 iff ( b4<=len b2 & b2. b4= b2. b3 & ( for b5 being oddNat holds ( b5<=len b2 & b2. b5= b2. b3 implies b5<= b4 ) ) ) ) ) & ( not ( not b3 is even & b3<=len b2 ) implies ( b4= b2.rfind b3 iff b4=len b2 ) ) );
Lemma33:
for b1 being _Graph for b2 being Walk of b1 for b3 being evenNat holds not ( b3indom b2 & ( for b4 being oddNat holds not ( b4= b3- 1 & b3- 1 indom b2 & b3+ 1 indom b2 & b2. b3Joins b2. b4,b2.(b3+ 1),b1 ) ) )
Lemma47:
for b1 being _Graph for b2, b3 being Walk of b1 for b4 being Nat holds not ( b4indom(b2.append b3) & not b4indom b2 & ( for b5 being Nat holds not ( b5<len b3 & b4=(len b2)+ b5 ) ) )
Lemma53:
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being Nat holds ( not b3 is even & b3<= b4 implies (b2.cut 1,b4).cut 1,b3= b2.cut 1,b3 )
Lemma56:
for b1 being _Graph for b2 being Walk of b1 for b3 being oddNat for b4 being Nat holds ( b4indom(b2.cut 1,b3) & b3<=len b2 implies (b2.cut 1,b3). b4= b2. b4 )
Lemma57:
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being oddNat holds ( b3<= b4 & b4<=len b2 & b2. b3= b2. b4 implies (len(b2.remove b3,b4))+ b4=(len b2)+ b3 )
Lemma62:
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being set for b5, b6 being oddNat holds ( b5<= b6 & b6<=len b2 & b2. b5= b2. b6 implies for b7 being Nat holds ( b7inSeg b5 implies (b2.remove b5,b6). b7= b2. b7 ) )
Lemma64:
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being oddNat holds ( b3<= b4 & b4<=len b2 & b2. b3= b2. b4 implies len(b2.remove b3,b4)=((len b2)+ b3)- b4 )
Lemma79:
for b1 being _Graph for b2 being Walk of b1 for b3 being set holds ( b3in b2.edges() iff ex b4 being evenNat st ( 1 <= b4 & b4<=len b2 & b2. b4= b3 ) )
Lemma80:
for b1 being _Graph for b2 being Walk of b1 for b3 being set holds not ( b3in b2.edges() & ( for b4, b5 being Vertex of b1 for b6 being oddNat holds not ( b6+ 2 <=len b2 & b4= b2. b6 & b3= b2.(b6+ 1) & b5= b2.(b6+ 2) & b3Joins b4,b5,b1 ) ) )
Lemma85:
for b1 being _Graph for b2 being Walk of b1 holds ( b2 is directed iff for b3 being oddNat holds ( b3<len b2 implies b2.(b3+ 1)DJoins b2. b3,b2.(b3+ 2),b1 ) )
Lemma91:
for b1 being _Graph for b2 being Walk of b1 holds ( b2 is Trail-like iff for b3, b4 being evenNat holds not ( 1 <= b3 & b3< b4 & b4<=len b2 & not b2. b3<> b2. b4 ) )
Lemma100:
for b1 being _Graph for b2 being Walk of b1 holds ( ( for b3, b4 being oddNat holds ( b3<=len b2 & b4<=len b2 & b2. b3= b2. b4 implies b3= b4 ) ) implies b2 is Path-like )
Lemma101:
for b1 being _Graph for b2 being Walk of b1 holds ( ( for b3 being oddNat holds ( b3<=len b2 implies b2.rfind b3= b3 ) ) implies b2 is Path-like )
for b1 being _Graph for b2 being Walk of b1 for b3 being oddNat holds ( b3<len b2 implies b1.walkOf(b2. b3),(b2.(b3+ 1)),(b2.(b3+ 2))= b2.cut b3,(b3+ 2) )
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being oddNat for b5 being Nat holds ( b3<= b4 & b4<=len b2 & b5indom(b2.cut b3,b4) implies ( (b2.cut b3,b4). b5= b2.((b3+ b5)- 1) & (b3+ b5)- 1 indom b2 ) )
for b1 being _Graph for b2 being Walk of b1 for b3, b4 being oddNat holds ( b3<= b4 & b4<=len b2 & b2. b3= b2. b4 implies for b5 being Nat holds ( b5inSeg b3 implies (b2.remove b3,b4). b5= b2. b5 ) ) by Lemma62;
for b1 being _Graph for b2 being Walk of b1 for b3 being set holds not ( b3in b2.edges() & ( for b4, b5 being Vertex of b1 for b6 being oddNat holds not ( b6+ 2 <=len b2 & b4= b2. b6 & b3= b2.(b6+ 1) & b5= b2.(b6+ 2) & b3Joins b4,b5,b1 ) ) ) by Lemma80;
for b1 being _Graph for b2 being Path of b1 holds ( not b2 is closed implies for b3, b4 being oddNat holds not ( b3< b4 & b4<=len b2 & not b2. b3<> b2. b4 ) )
for b1 being _Graph for b2, b3 being Walk of b1 holds ( b2 is Subwalk of b3 implies for b4 being oddNat holds not ( b4<=len b2 & ( for b5 being oddNat holds not ( b4<= b5 & b5<=len b3 & b2. b4= b3. b5 ) ) ) )
for b1 being _Graph for b2, b3 being Walk of b1 holds ( b2 is Subwalk of b3 implies for b4 being evenNat holds not ( 1 <= b4 & b4<=len b2 & ( for b5 being evenNat holds not ( b4<= b5 & b5<=len b3 & b2. b4= b3. b5 ) ) ) )