Article JORDAN5B, MML version 4.99.1005

:: JORDAN5B:th 1
theorem
for b1 being natural set
      st 1 <= b1
   holds b1 -' 1 < b1;

:: JORDAN5B:th 2
theorem
for b1, b2 being natural set
      st b1 + 1 <= b2
   holds 1 <= b2 -' b1;

:: JORDAN5B:th 3
theorem
for b1, b2 being natural set
      st 1 <= b1 & 1 <= b2
   holds (b2 -' b1) + 1 <= b2;

:: JORDAN5B:th 4
theorem
for b1 being real set
      st b1 in the carrier of I[01]
   holds 1 - b1 in the carrier of I[01];

:: JORDAN5B:th 5
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `2 <> b2 `2 & b3 in LSeg(b1,b2) & b3 `2 = b1 `2
   holds b3 = b1;

:: JORDAN5B:th 6
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 `1 <> b2 `1 & b3 in LSeg(b1,b2) & b3 `1 = b1 `1
   holds b3 = b1;

:: JORDAN5B:th 7
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b2
for b4 being Element of NAT
      st 1 <= b4 & b4 + 1 <= len b1 & b1 is being_S-Seq & b2 = L~ b1 & b3 is being_homeomorphism(I[01], (TOP-REAL 2) | b2) & b3 . 0 = b1 /. 1 & b3 . 1 = b1 /. len b1
   holds ex b5, b6 being Element of REAL st
      b5 < b6 & 0 <= b5 & b5 <= 1 & 0 <= b6 & b6 <= 1 & LSeg(b1,b4) = b3 .: [.b5,b6.] & b3 . b5 = b1 /. b4 & b3 . b6 = b1 /. (b4 + 1);

:: JORDAN5B:th 8
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being non empty Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b2
for b5 being Element of NAT
for b6 being non empty Element of bool the carrier of I[01]
      st b1 is being_S-Seq & b4 is being_homeomorphism(I[01], (TOP-REAL 2) | b2) & b4 . 0 = b1 /. 1 & b4 . 1 = b1 /. len b1 & 1 <= b5 & b5 + 1 <= len b1 & b4 .: b6 = LSeg(b1,b5) & b2 = L~ b1 & b3 = LSeg(b1,b5)
   holds ex b7 being Function-like quasi_total Relation of the carrier of I[01] | b6,the carrier of (TOP-REAL 2) | b3 st
      b7 = b4 | b6 &
       b7 is being_homeomorphism(I[01] | b6, (TOP-REAL 2) | b3);

:: JORDAN5B:th 9
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 & b3 in LSeg(b1,b2)
   holds LE b3,b3,b1,b2;

:: JORDAN5B:th 10
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 <> b3 & b1 in LSeg(b2,b3)
   holds LE b2,b1,b2,b3;

:: JORDAN5B:th 11
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 in LSeg(b2,b3) & b2 <> b3
   holds LE b1,b3,b2,b3;

:: JORDAN5B:th 12
theorem
for b1, b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 & LE b3,b4,b1,b2 & LE b4,b5,b1,b2
   holds LE b3,b5,b1,b2;

:: JORDAN5B:th 13
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 <> b2
   holds LSeg(b1,b2) = {b3 where b3 is Element of the carrier of TOP-REAL 2: LE b1,b3,b1,b2 & LE b3,b2,b1,b2};

:: JORDAN5B:th 14
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3, b4 being Element of the carrier of TOP-REAL b1
      st b2 is_an_arc_of b3,b4
   holds b2 is_an_arc_of b4,b3;

:: JORDAN5B:th 15
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of TOP-REAL 2
      st b2 is being_S-Seq & 1 <= b1 & b1 + 1 <= len b2 & b3 = LSeg(b2,b1)
   holds b3 is_an_arc_of b2 /. b1,b2 /. (b1 + 1);

:: JORDAN5B:th 16
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 &
         b2 <= len b1 &
         b1 is being_S-Seq &
         b1 /. 1 in L~ mid(b1,b2,len b1)
   holds b2 = 1;

:: JORDAN5B:th 17
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b1 is being_S-Seq & b2 = b1 . len b1
   holds L_Cut(b1,b2) = <*b2*>;

:: JORDAN5B:th 21
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b2 <> b1 . len b1 & b1 is being_S-Seq
   holds Index(b2,L_Cut(b1,b2)) = 1;

:: JORDAN5B:th 22
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b1 is being_S-Seq & b2 <> b1 . len b1
   holds b2 in L~ L_Cut(b1,b2);

:: JORDAN5B:th 23
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b1 is being_S-Seq & b2 <> b1 . 1
   holds b2 in L~ R_Cut(b1,b2);

:: JORDAN5B:th 24
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b1 is one-to-one
   holds B_Cut(b1,b2,b2) = <*b2*>;

:: JORDAN5B:th 25
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b3 in L~ b1 & b3 <> b1 . len b1 & b2 = b1 . len b1 & b1 is being_S-Seq
   holds b2 in L~ L_Cut(b1,b3);

:: JORDAN5B:th 26
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 <> b1 . len b1 & b2 in L~ b1 & b3 in L~ b1 & b1 is being_S-Seq & not b2 in L~ L_Cut(b1,b3)
   holds b3 in L~ L_Cut(b1,b2);

:: JORDAN5B:th 27
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 & b3 in L~ b1 & b1 is being_S-Seq
   holds L~ B_Cut(b1,b2,b3) c= L~ b1;

:: JORDAN5B:th 28
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b3 <= len GoB b1 & b2 < b3
   holds (LSeg((GoB b1) *(1,width GoB b1),(GoB b1) *(b2,width GoB b1))) /\ LSeg((GoB b1) *(b3,width GoB b1),(GoB b1) *(len GoB b1,width GoB b1)) = {};

:: JORDAN5B:th 29
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st 1 <= b2 & b3 <= width GoB b1 & b2 < b3
   holds (LSeg((GoB b1) *(len GoB b1,1),(GoB b1) *(len GoB b1,b2))) /\ LSeg((GoB b1) *(len GoB b1,b3),(GoB b1) *(len GoB b1,width GoB b1)) = {};

:: JORDAN5B:th 30
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is being_S-Seq
   holds L_Cut(b1,b1 /. 1) = b1;

:: JORDAN5B:th 31
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
      st b1 is being_S-Seq
   holds R_Cut(b1,b1 /. len b1) = b1;

:: JORDAN5B:th 32
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1
   holds b2 in LSeg(b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1));

:: JORDAN5B:th 33
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st b1 is unfolded & b1 is s.n.c. & b1 is one-to-one & 2 <= len b1 & b1 /. 1 in LSeg(b1,b2)
   holds b2 = 1;

:: JORDAN5B:th 34
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
for b3 being Element of bool the carrier of TOP-REAL 2
      st 1 <= b2 &
         b2 <= width GoB b1 &
         b3 = LSeg((GoB b1) *(1,b2),(GoB b1) *(len GoB b1,b2))
   holds b3 is_S-P_arc_joining (GoB b1) *(1,b2),(GoB b1) *(len GoB b1,b2);

:: JORDAN5B:th 35
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
for b3 being Element of bool the carrier of TOP-REAL 2
      st 1 <= b2 &
         b2 <= len GoB b1 &
         b3 = LSeg((GoB b1) *(b2,1),(GoB b1) *(b2,width GoB b1))
   holds b3 is_S-P_arc_joining (GoB b1) *(b2,1),(GoB b1) *(b2,width GoB b1);

:: JORDAN5B:th 36
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
      st b2 in L~ b1 &
         b3 in L~ b1 &
         (Index(b3,b1) <= Index(b2,b1) implies Index(b2,b1) = Index(b3,b1) &
          LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)) &
         b2 <> b3
   holds L~ B_Cut(b1,b2,b3) c= L~ b1;