Article JORDAN1B, MML version 4.99.1005

:: JORDAN1B:th 2
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
      b1 is empty
   iff
      Rev b1 is empty;

:: JORDAN1B:th 3
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2
   holds mid(b2,b3,b4) is not empty;

:: JORDAN1B:th 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st 1 <= len b1 & b2 in L~ b1
   holds (R_Cut(b1,b2)) . 1 = b1 . 1;

:: JORDAN1B:th 5
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 in L~ b1
   holds (L_Cut(b1,b2)) . len L_Cut(b1,b2) = b1 . len b1;

:: JORDAN1B:th 6
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   W-max b1 <> E-max b1;

:: JORDAN1B:th 7
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
      st 1 <= b1 & b1 < len b3
   holds (mid(b3,b1,(len b3) -' 1)) ^ <*b3 /. len b3*> = mid(b3,b1,len b3);

:: JORDAN1B:th 8
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 & LSeg(b1,b2) is vertical
   holds <*b1,b2*> is being_S-Seq;

:: JORDAN1B:th 9
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 & LSeg(b1,b2) is horizontal
   holds <*b1,b2*> is being_S-Seq;

:: JORDAN1B:th 10
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
      st b1 is_in_the_area_of b2
   holds Rotate(b1,b3) is_in_the_area_of b2;

:: JORDAN1B:th 11
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   Rotate(b1,b2) is_in_the_area_of b1;

:: JORDAN1B:th 12
theorem
for b1 being Relation-like Function-like FinSequence-like set holds
   1 <= Center b1;

:: JORDAN1B:th 13
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st 1 <= len b1
   holds Center b1 <= len b1;

:: JORDAN1B:th 14
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) * holds
   Center b1 <= len b1;

:: JORDAN1B:th 15
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st 2 <= len b1
   holds 1 < Center b1;

:: JORDAN1B:th 16
theorem
for b1 being Relation-like Function-like FinSequence-like set
      st 3 <= len b1
   holds Center b1 < len b1;

:: JORDAN1B:th 17
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
   Center Gauge(b1,b2) = (2 |^ (b2 -' 1)) + 2;

:: JORDAN1B:th 18
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   b1 c= cell(Gauge(b1,0),2,2);

:: JORDAN1B:th 19
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   not cell(Gauge(b1,0),2,2) c= BDD b1;

:: JORDAN1B:th 20
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (Gauge(b1,1)) *(Center Gauge(b1,1),1) = |[((W-bound b1) + E-bound b1) / 2,S-bound L~ Cage(b1,1)]|;

:: JORDAN1B:th 21
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   (Gauge(b1,1)) *(Center Gauge(b1,1),len Gauge(b1,1)) = |[((W-bound b1) + E-bound b1) / 2,N-bound L~ Cage(b1,1)]|;

:: JORDAN1B:th 22
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, b3, b4 being Element of NAT
for b5 being Element of the carrier of TOP-REAL 2
      st 1 <= b2 & b2 < width b1 & 1 <= b3 & b3 <= len b1 & 1 <= b4 & b4 <= width b1 & b5 in cell(b1,len b1,b2) & b5 `1 = (b1 *(b3,b4)) `1
   holds len b1 = b3;

:: JORDAN1B:th 23
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, b3, b4, b5 being Element of NAT
for b6 being Element of the carrier of TOP-REAL 2
      st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 < width b1 & 1 <= b4 & b4 <= len b1 & 1 <= b5 & b5 <= width b1 & b6 in cell(b1,b2,b3) & b6 `1 = (b1 *(b4,b5)) `1 & b2 <> b4
   holds b2 = b4 -' 1;

:: JORDAN1B:th 24
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, b3, b4 being Element of NAT
for b5 being Element of the carrier of TOP-REAL 2
      st 1 <= b2 & b2 < len b1 & 1 <= b3 & b3 <= len b1 & 1 <= b4 & b4 <= width b1 & b5 in cell(b1,b2,width b1) & b5 `2 = (b1 *(b3,b4)) `2
   holds width b1 = b4;

:: JORDAN1B:th 25
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, b3, b4, b5 being Element of NAT
for b6 being Element of the carrier of TOP-REAL 2
      st 1 <= b2 & b2 < len b1 & 1 <= b3 & b3 <= width b1 & 1 <= b4 & b4 <= len b1 & 1 <= b5 & b5 <= width b1 & b6 in cell(b1,b2,b3) & b6 `2 = (b1 *(b4,b5)) `2 & b3 <> b5
   holds b3 = b5 -' 1;

:: JORDAN1B:th 28
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 < b3 & b3 < len Gauge(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,1),(Gauge(b2,b1)) *(b3,len Gauge(b2,b1))) meets Upper_Arc b2;

:: JORDAN1B:th 29
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
      st 1 < b3 & b3 < len Gauge(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,1),(Gauge(b2,b1)) *(b3,len Gauge(b2,b1))) meets Lower_Arc b2;

:: JORDAN1B:th 30
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   LSeg((Gauge(b2,b1)) *(Center Gauge(b2,b1),1),(Gauge(b2,b1)) *(Center Gauge(b2,b1),len Gauge(b2,b1))) meets Upper_Arc b2;

:: JORDAN1B:th 31
theorem
for b1 being Element of NAT
for b2 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
   LSeg((Gauge(b2,b1)) *(Center Gauge(b2,b1),1),(Gauge(b2,b1)) *(Center Gauge(b2,b1),len Gauge(b2,b1))) meets Lower_Arc b2;

:: JORDAN1B:th 32
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
for b3 being Element of NAT
      st 1 <= b3 & b3 <= len Gauge(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,1),(Gauge(b2,b1)) *(b3,len Gauge(b2,b1))) meets Upper_Arc L~ Cage(b2,b1);

:: JORDAN1B:th 33
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
for b3 being Element of NAT
      st 1 <= b3 & b3 <= len Gauge(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,1),(Gauge(b2,b1)) *(b3,len Gauge(b2,b1))) meets Lower_Arc L~ Cage(b2,b1);

:: JORDAN1B:th 34
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
   LSeg((Gauge(b2,b1)) *(Center Gauge(b2,b1),1),(Gauge(b2,b1)) *(Center Gauge(b2,b1),len Gauge(b2,b1))) meets Upper_Arc L~ Cage(b2,b1);

:: JORDAN1B:th 35
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
   LSeg((Gauge(b2,b1)) *(Center Gauge(b2,b1),1),(Gauge(b2,b1)) *(Center Gauge(b2,b1),len Gauge(b2,b1))) meets Lower_Arc L~ Cage(b2,b1);

:: JORDAN1B:th 36
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 NAT
      st b2 <= width b1
   holds cell(b1,0,b2) is not Bounded(2);

:: JORDAN1B:th 37
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 NAT
      st b2 <= width b1
   holds cell(b1,len b1,b2) is not Bounded(2);

:: JORDAN1B:th 38
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b2 <= width Gauge(b1,b3)
   holds cell(Gauge(b1,b3),0,b2) c= UBD b1;

:: JORDAN1B:th 39
theorem
for b1 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
      st b2 <= len Gauge(b1,b3)
   holds cell(Gauge(b1,b3),len Gauge(b1,b3),b2) c= UBD b1;

:: JORDAN1B:th 40
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st b2 <= len Gauge(b1,b3) & b4 <= width Gauge(b1,b3) & cell(Gauge(b1,b3),b2,b4) c= BDD b1
   holds 1 < b4;

:: JORDAN1B:th 41
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st b2 <= len Gauge(b1,b3) & b4 <= width Gauge(b1,b3) & cell(Gauge(b1,b3),b2,b4) c= BDD b1
   holds 1 < b2;

:: JORDAN1B:th 42
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st b2 <= len Gauge(b1,b3) & b4 <= width Gauge(b1,b3) & cell(Gauge(b1,b3),b2,b4) c= BDD b1
   holds b4 + 1 < width Gauge(b1,b3);

:: JORDAN1B:th 43
theorem
for b1 being connected compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of NAT
      st b2 <= len Gauge(b1,b3) & b4 <= width Gauge(b1,b3) & cell(Gauge(b1,b3),b2,b4) c= BDD b1
   holds b2 + 1 < len Gauge(b1,b3);

:: JORDAN1B:th 44
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
      st ex b3, b4 being Element of NAT st
           b3 <= len Gauge(b1,b2) & b4 <= width Gauge(b1,b2) & cell(Gauge(b1,b2),b3,b4) c= BDD b1
   holds 1 <= b2;