Article JORDAN1F, MML version 4.99.1005

:: JORDAN1F:th 1
theorem
for b1, b2, b3 being Element of NAT
for b4 being FinSequence of the carrier of TOP-REAL 2
for b5 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 b4 is_sequence_on b5 &
         LSeg(b5 *(b1,b2),b5 *(b1,b3)) meets L~ b4 &
         [b1,b2] in Indices b5 &
         [b1,b3] in Indices b5 &
         b2 <= b3
   holds ex b6 being Element of NAT st
      b2 <= b6 &
       b6 <= b3 &
       (b5 *(b1,b6)) `2 = inf (proj2 .: ((LSeg(b5 *(b1,b2),b5 *(b1,b3))) /\ L~ b4));

:: JORDAN1F:th 2
theorem
for b1, b2, b3 being Element of NAT
for b4 being FinSequence of the carrier of TOP-REAL 2
for b5 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 b4 is_sequence_on b5 &
         LSeg(b5 *(b1,b2),b5 *(b1,b3)) meets L~ b4 &
         [b1,b2] in Indices b5 &
         [b1,b3] in Indices b5 &
         b2 <= b3
   holds ex b6 being Element of NAT st
      b2 <= b6 &
       b6 <= b3 &
       (b5 *(b1,b6)) `2 = sup (proj2 .: ((LSeg(b5 *(b1,b2),b5 *(b1,b3))) /\ L~ b4));

:: JORDAN1F:th 3
theorem
for b1, b2, b3 being Element of NAT
for b4 being FinSequence of the carrier of TOP-REAL 2
for b5 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 b4 is_sequence_on b5 &
         LSeg(b5 *(b1,b2),b5 *(b3,b2)) meets L~ b4 &
         [b1,b2] in Indices b5 &
         [b3,b2] in Indices b5 &
         b1 <= b3
   holds ex b6 being Element of NAT st
      b1 <= b6 &
       b6 <= b3 &
       (b5 *(b6,b2)) `1 = inf (proj1 .: ((LSeg(b5 *(b1,b2),b5 *(b3,b2))) /\ L~ b4));

:: JORDAN1F:th 4
theorem
for b1, b2, b3 being Element of NAT
for b4 being FinSequence of the carrier of TOP-REAL 2
for b5 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 b4 is_sequence_on b5 &
         LSeg(b5 *(b1,b2),b5 *(b3,b2)) meets L~ b4 &
         [b1,b2] in Indices b5 &
         [b3,b2] in Indices b5 &
         b1 <= b3
   holds ex b6 being Element of NAT st
      b1 <= b6 &
       b6 <= b3 &
       (b5 *(b6,b2)) `1 = sup (proj1 .: ((LSeg(b5 *(b1,b2),b5 *(b3,b2))) /\ L~ b4));

:: JORDAN1F:th 5
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (Upper_Seq(b1,b2)) /. 1 = W-min L~ Cage(b1,b2);

:: JORDAN1F:th 6
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (Lower_Seq(b1,b2)) /. 1 = E-max L~ Cage(b1,b2);

:: JORDAN1F:th 7
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (Upper_Seq(b1,b2)) /. len Upper_Seq(b1,b2) = E-max L~ Cage(b1,b2);

:: JORDAN1F:th 8
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   (Lower_Seq(b1,b2)) /. len Lower_Seq(b1,b2) = W-min L~ Cage(b1,b2);

:: JORDAN1F:th 9
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT
      st (L~ Upper_Seq(b1,b2) = Upper_Arc L~ Cage(b1,b2) implies L~ Lower_Seq(b1,b2) <> Lower_Arc L~ Cage(b1,b2))
   holds L~ Upper_Seq(b1,b2) = Lower_Arc L~ Cage(b1,b2) &
    L~ Lower_Seq(b1,b2) = Upper_Arc L~ Cage(b1,b2);

:: JORDAN1F:th 10
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Upper_Seq(b1,b2) is_sequence_on Gauge(b1,b2);

:: JORDAN1F:th 11
theorem
for b1 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 b2 being Element of the carrier of TOP-REAL 2
for b3 being FinSequence of the carrier of TOP-REAL 2
      st b3 is_sequence_on b1 &
         (ex b4, b5 being Element of NAT st
            [b4,b5] in Indices b1 & b2 = b1 *(b4,b5)) &
         (for b4, b5, b6, b7 being Element of NAT
               st [b4,b5] in Indices b1 & [b6,b7] in Indices b1 & b2 = b1 *(b4,b5) & b3 /. 1 = b1 *(b6,b7)
            holds (abs (b6 - b4)) + abs (b7 - b5) = 1)
   holds <*b2*> ^ b3 is_sequence_on b1;

:: JORDAN1F:th 12
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being Element of NAT holds
   Lower_Seq(b1,b2) is_sequence_on Gauge(b1,b2);

:: JORDAN1F:th 13
theorem
for b1 being Element of NAT
for b2 being non empty being_simple_closed_curve compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b3 `1 = ((W-bound b2) + E-bound b2) / 2 &
         b3 `2 = inf (proj2 .: ((LSeg((Gauge(b2,1)) *(Center Gauge(b2,1),1),(Gauge(b2,1)) *(Center Gauge(b2,1),width Gauge(b2,1)))) /\ Upper_Arc L~ Cage(b2,b1 + 1)))
   holds ex b4 being Element of NAT st
      1 <= b4 &
       b4 <= width Gauge(b2,b1 + 1) &
       b3 = (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4);