Article JORDAN15, MML version 4.99.1005

:: JORDAN15:th 1
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
      st b1 meets b2
   holds proj1 .: b1 meets proj1 .: b2;

:: JORDAN15:th 2
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being real set
      st b1 misses b2 & b1 c= Horizontal_Line b3 & b2 c= Horizontal_Line b3
   holds proj1 .: b1 misses proj1 .: b2;

:: JORDAN15:th 3
theorem
for b1 being closed Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds proj1 .: b1 is closed;

:: JORDAN15:th 4
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
   proj1 .: b1 is compact;

:: JORDAN15:th 7
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, b6 being Element of NAT
      st 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= b5 & b5 <= b6 & b6 <= b4 & b4 <= width b1
   holds LSeg(b1 *(b2,b5),b1 *(b2,b6)) c= LSeg(b1 *(b2,b3),b1 *(b2,b4));

:: JORDAN15:th 8
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, b6 being Element of NAT
      st 1 <= b2 & b2 <= width b1 & 1 <= b3 & b3 <= b5 & b5 <= b6 & b6 <= b4 & b4 <= len b1
   holds LSeg(b1 *(b5,b2),b1 *(b6,b2)) c= LSeg(b1 *(b3,b2),b1 *(b4,b2));

:: JORDAN15:th 9
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
      st 1 <= b2 & b2 <= b4 & b4 <= b5 & b5 <= b3 & b3 <= width b1
   holds LSeg(b1 *(Center b1,b4),b1 *(Center b1,b5)) c= LSeg(b1 *(Center b1,b2),b1 *(Center b1,b3));

:: JORDAN15:th 10
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) *
   st len b1 = width b1
for b2, b3, b4, b5 being Element of NAT
      st 1 <= b2 & b2 <= b4 & b4 <= b5 & b5 <= b3 & b3 <= len b1
   holds LSeg(b1 *(b4,Center b1),b1 *(b5,Center b1)) c= LSeg(b1 *(b2,Center b1),b1 *(b3,Center b1));

:: JORDAN15:th 11
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Lower_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)};

:: JORDAN15:th 12
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Upper_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)};

:: JORDAN15:th 13
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Lower_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Upper_Seq(b2,b1)
   holds ex b6, b7 being Element of NAT st
      b4 <= b6 &
       b6 <= b7 &
       b7 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b7))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)} &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b7))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b7)};

:: JORDAN15:th 14
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Lower_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)};

:: JORDAN15:th 15
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Upper_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b6,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)};

:: JORDAN15:th 16
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Lower_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Upper_Seq(b2,b1)
   holds ex b6, b7 being Element of NAT st
      b4 <= b6 &
       b6 <= b7 &
       b7 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b7,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)} &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b7,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b7,b3)};

:: JORDAN15:th 17
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Upper_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)};

:: JORDAN15:th 18
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Lower_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)};

:: JORDAN15:th 19
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, b4, b5 being Element of NAT
      st 1 <= b3 &
         b3 <= len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Lower_Seq(b2,b1)
   holds ex b6, b7 being Element of NAT st
      b4 <= b6 &
       b6 <= b7 &
       b7 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b7))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b6)} &
       (LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b3,b7))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b7)};

:: JORDAN15:th 20
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Upper_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)};

:: JORDAN15:th 21
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Lower_Seq(b2,b1)
   holds ex b6 being Element of NAT st
      b4 <= b6 &
       b6 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b6,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)};

:: JORDAN15:th 22
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, b4, b5 being Element of NAT
      st 1 <= b4 &
         b4 <= b5 &
         b5 <= len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Lower_Seq(b2,b1)
   holds ex b6, b7 being Element of NAT st
      b4 <= b6 &
       b6 <= b7 &
       b7 <= b5 &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b7,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b6,b3)} &
       (LSeg((Gauge(b2,b1)) *(b6,b3),(Gauge(b2,b1)) *(b7,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b7,b3)};

:: JORDAN15:th 23
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN15:th 24
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b3,b5) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN15:th 25
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b3,b5) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN15:th 26
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b3,b5) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b3,b4) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN15:th 27
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, b4 being Element of NAT
      st 1 <= b3 &
         b3 <= b4 &
         b4 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4)) meets Lower_Arc b2;

:: JORDAN15: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, b4 being Element of NAT
      st 1 <= b3 &
         b3 <= b4 &
         b4 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4)) meets Upper_Arc b2;

:: JORDAN15:th 29
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Lower_Seq(b2,b1)
   holds b4 <> b5;

:: JORDAN15: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
for b3, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b5,b3)} &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b3)}
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15: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
for b3, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b5,b3)} &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b3)}
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 32
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15:th 33
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 34
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b5,b3) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15:th 35
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b5,b3) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 36
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, b4 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1)) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)),(Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1))) meets Lower_Arc b2;

:: JORDAN15:th 37
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, b4 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1)) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)),(Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1))) meets Upper_Arc b2;

:: JORDAN15:th 38
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b3)} &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b5,b3)}
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15:th 39
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b3)} &
         (LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b5,b3)}
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 40
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15:th 41
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         (Gauge(b2,b1)) *(b4,b3) in L~ Upper_Seq(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in L~ Lower_Seq(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 42
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b4,b3) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Lower_Arc b2;

:: JORDAN15:th 43
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, b4, b5 being Element of NAT
      st 1 < b4 &
         b4 <= b5 &
         b5 < len Gauge(b2,b1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1) &
         0 < b1 &
         (Gauge(b2,b1)) *(b4,b3) in Upper_Arc L~ Cage(b2,b1) &
         (Gauge(b2,b1)) *(b5,b3) in Lower_Arc L~ Cage(b2,b1)
   holds LSeg((Gauge(b2,b1)) *(b4,b3),(Gauge(b2,b1)) *(b5,b3)) meets Upper_Arc b2;

:: JORDAN15:th 44
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, b4 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1)) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)),(Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1))) meets Lower_Arc b2;

:: JORDAN15:th 45
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, b4 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1)) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds LSeg((Gauge(b2,b1 + 1)) *(b3,Center Gauge(b2,b1 + 1)),(Gauge(b2,b1 + 1)) *(b4,Center Gauge(b2,b1 + 1))) meets Upper_Arc b2;

:: JORDAN15:th 46
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, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Upper_Arc b2;

:: JORDAN15:th 47
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, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 <= b4 &
         b4 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Lower_Arc b2;

:: JORDAN15:th 48
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, b4, b5, b6 being Element of NAT
      st 1 < b4 &
         b4 <= b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Upper_Arc b2;

:: JORDAN15:th 49
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, b4, b5, b6 being Element of NAT
      st 1 < b4 &
         b4 <= b3 &
         b3 < len Gauge(b2,b1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1) &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b4,b6)} &
         ((LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)}
   holds (LSeg((Gauge(b2,b1)) *(b3,b5),(Gauge(b2,b1)) *(b3,b6))) \/ LSeg((Gauge(b2,b1)) *(b3,b6),(Gauge(b2,b1)) *(b4,b6)) meets Lower_Arc b2;

:: JORDAN15:th 50
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, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 < b4 &
         b4 < len Gauge(b2,b1 + 1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b6) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,b5) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(b4,b5),(Gauge(b2,b1 + 1)) *(b4,b6))) \/ LSeg((Gauge(b2,b1 + 1)) *(b4,b6),(Gauge(b2,b1 + 1)) *(b3,b6)) meets Upper_Arc b2;

:: JORDAN15:th 51
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, b4, b5, b6 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 < b4 &
         b4 < len Gauge(b2,b1 + 1) &
         1 <= b5 &
         b5 <= b6 &
         b6 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b6) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b4,b5) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(b4,b5),(Gauge(b2,b1 + 1)) *(b4,b6))) \/ LSeg((Gauge(b2,b1 + 1)) *(b4,b6),(Gauge(b2,b1 + 1)) *(b3,b6)) meets Lower_Arc b2;

:: JORDAN15:th 52
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b5) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b5))) \/ LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b5),(Gauge(b2,b1 + 1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN15:th 53
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, b4, b5 being Element of NAT
      st 1 < b3 &
         b3 < len Gauge(b2,b1 + 1) &
         1 <= b4 &
         b4 <= b5 &
         b5 <= width Gauge(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(b3,b5) in Upper_Arc L~ Cage(b2,b1 + 1) &
         (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4) in Lower_Arc L~ Cage(b2,b1 + 1)
   holds (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b5))) \/ LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b5),(Gauge(b2,b1 + 1)) *(b3,b5)) meets Lower_Arc b2;