Article JORDAN1I, MML version 4.99.1005

:: JORDAN1I:th 3
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   1 < (W-min L~ Cage(b2,b1)) .. Cage(b2,b1);

:: JORDAN1I:th 4
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   1 < (E-max L~ Cage(b2,b1)) .. Cage(b2,b1);

:: JORDAN1I:th 5
theorem
for b1 being Element of NAT
for b2 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   1 < (S-max L~ Cage(b2,b1)) .. Cage(b2,b1);

:: JORDAN1I:th 6
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 the carrier of TOP-REAL 2
      st b2 in rng b1
   holds left_cell(b1,b2 .. b1) = left_cell(Rotate(b1,b2),1);

:: JORDAN1I:th 7
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 the carrier of TOP-REAL 2
      st b2 in rng b1
   holds right_cell(b1,b2 .. b1) = right_cell(Rotate(b1,b2),1);

:: JORDAN1I:th 8
theorem
for b1 being Element of NAT
for b2 being non empty connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   W-min b2 in right_cell(Rotate(Cage(b2,b1),W-min L~ Cage(b2,b1)),1);

:: JORDAN1I:th 9
theorem
for b1 being Element of NAT
for b2 being non empty connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   E-max b2 in right_cell(Rotate(Cage(b2,b1),E-max L~ Cage(b2,b1)),1);

:: JORDAN1I:th 10
theorem
for b1 being Element of NAT
for b2 being non empty connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   S-max b2 in right_cell(Rotate(Cage(b2,b1),S-max L~ Cage(b2,b1)),1);

:: JORDAN1I:th 11
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 `1 < W-bound L~ b1
   holds b2 in LeftComp b1;

:: JORDAN1I:th 12
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st E-bound L~ b1 < b2 `1
   holds b2 in LeftComp b1;

:: JORDAN1I:th 13
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 `2 < S-bound L~ b1
   holds b2 in LeftComp b1;

:: JORDAN1I:th 14
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st N-bound L~ b1 < b2 `2
   holds b2 in LeftComp b1;

:: JORDAN1I:th 15
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 & b5 + 1 <= len b1 & [b3,b4] in Indices b2 & [b3 + 1,b4] in Indices b2 & b1 /. b5 = b2 *(b3 + 1,b4) & b1 /. (b5 + 1) = b2 *(b3,b4)
   holds b4 < width b2;

:: JORDAN1I:th 16
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 &
         b5 + 1 <= len b1 &
         [b3,b4] in Indices b2 &
         [b3,b4 + 1] in Indices b2 &
         b1 /. b5 = b2 *(b3,b4) &
         b1 /. (b5 + 1) = b2 *(b3,b4 + 1)
   holds b3 < len b2;

:: JORDAN1I:th 17
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 &
         b5 + 1 <= len b1 &
         [b3,b4] in Indices b2 &
         [b3 + 1,b4] in Indices b2 &
         b1 /. b5 = b2 *(b3,b4) &
         b1 /. (b5 + 1) = b2 *(b3 + 1,b4)
   holds 1 < b4;

:: JORDAN1I:th 18
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 & b5 + 1 <= len b1 & [b3,b4] in Indices b2 & [b3,b4 + 1] in Indices b2 & b1 /. b5 = b2 *(b3,b4 + 1) & b1 /. (b5 + 1) = b2 *(b3,b4)
   holds 1 < b3;

:: JORDAN1I:th 19
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 & b5 + 1 <= len b1 & [b3,b4] in Indices b2 & [b3 + 1,b4] in Indices b2 & b1 /. b5 = b2 *(b3 + 1,b4) & b1 /. (b5 + 1) = b2 *(b3,b4)
   holds (b1 /. b5) `2 <> N-bound L~ b1;

:: JORDAN1I:th 20
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 &
         b5 + 1 <= len b1 &
         [b3,b4] in Indices b2 &
         [b3,b4 + 1] in Indices b2 &
         b1 /. b5 = b2 *(b3,b4) &
         b1 /. (b5 + 1) = b2 *(b3,b4 + 1)
   holds (b1 /. b5) `1 <> E-bound L~ b1;

:: JORDAN1I:th 21
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 &
         b5 + 1 <= len b1 &
         [b3,b4] in Indices b2 &
         [b3 + 1,b4] in Indices b2 &
         b1 /. b5 = b2 *(b3,b4) &
         b1 /. (b5 + 1) = b2 *(b3 + 1,b4)
   holds (b1 /. b5) `2 <> S-bound L~ b1;

:: JORDAN1I:th 22
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
   st b1 is_sequence_on b2
for b3, b4, b5 being Element of NAT
      st 1 <= b5 & b5 + 1 <= len b1 & [b3,b4] in Indices b2 & [b3,b4 + 1] in Indices b2 & b1 /. b5 = b2 *(b3,b4 + 1) & b1 /. (b5 + 1) = b2 *(b3,b4)
   holds (b1 /. b5) `1 <> W-bound L~ b1;

:: JORDAN1I:th 23
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b3 being Element of NAT
      st b1 is_sequence_on b2 & 1 <= b3 & b3 + 1 <= len b1 & b1 /. b3 = W-min L~ b1
   holds ex b4, b5 being Element of NAT st
      [b4,b5] in Indices b2 &
       [b4,b5 + 1] in Indices b2 &
       b1 /. b3 = b2 *(b4,b5) &
       b1 /. (b3 + 1) = b2 *(b4,b5 + 1);

:: JORDAN1I:th 24
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b3 being Element of NAT
      st b1 is_sequence_on b2 & 1 <= b3 & b3 + 1 <= len b1 & b1 /. b3 = N-min L~ b1
   holds ex b4, b5 being Element of NAT st
      [b4,b5] in Indices b2 &
       [b4 + 1,b5] in Indices b2 &
       b1 /. b3 = b2 *(b4,b5) &
       b1 /. (b3 + 1) = b2 *(b4 + 1,b5);

:: JORDAN1I:th 25
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b3 being Element of NAT
      st b1 is_sequence_on b2 & 1 <= b3 & b3 + 1 <= len b1 & b1 /. b3 = E-max L~ b1
   holds ex b4, b5 being Element of NAT st
      [b4,b5 + 1] in Indices b2 & [b4,b5] in Indices b2 & b1 /. b3 = b2 *(b4,b5 + 1) & b1 /. (b3 + 1) = b2 *(b4,b5);

:: JORDAN1I:th 26
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
for b2 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b3 being Element of NAT
      st b1 is_sequence_on b2 & 1 <= b3 & b3 + 1 <= len b1 & b1 /. b3 = S-max L~ b1
   holds ex b4, b5 being Element of NAT st
      [b4 + 1,b5] in Indices b2 & [b4,b5] in Indices b2 & b1 /. b3 = b2 *(b4 + 1,b5) & b1 /. (b3 + 1) = b2 *(b4,b5);

:: JORDAN1I:th 27
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
      b1 is clockwise_oriented
   iff
      (Rotate(b1,W-min L~ b1)) /. 2 in W-most L~ b1;

:: JORDAN1I: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 holds
      b1 is clockwise_oriented
   iff
      (Rotate(b1,E-max L~ b1)) /. 2 in E-most L~ b1;

:: JORDAN1I: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 holds
      b1 is clockwise_oriented
   iff
      (Rotate(b1,S-max L~ b1)) /. 2 in S-most L~ b1;

:: JORDAN1I:th 30
theorem
for b1, b2 being Element of NAT
for b3 being non empty being_simple_closed_curve compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b4 being Element of the carrier of TOP-REAL 2
      st b4 `1 = ((W-bound b3) + E-bound b3) / 2 &
         0 < b1 &
         1 <= b2 &
         b2 <= width Gauge(b3,b1) &
         (Gauge(b3,b1)) *(Center Gauge(b3,b1),b2) in Upper_Arc L~ Cage(b3,b1) &
         b4 `2 = sup (proj2 .: ((LSeg((Gauge(b3,1)) *(Center Gauge(b3,1),1),(Gauge(b3,b1)) *(Center Gauge(b3,b1),b2))) /\ Lower_Arc L~ Cage(b3,b1)))
   holds ex b5 being Element of NAT st
      1 <= b5 &
       b5 <= width Gauge(b3,b1) &
       b4 = (Gauge(b3,b1)) *(Center Gauge(b3,b1),b5);