Article JORDAN1J, MML version 4.99.1005

:: JORDAN1J:th 1
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 <= b4 & b4 <= width b1 & 1 <= b5 & b5 <= width b1 & 1 <= b2 & b2 < b3 & b3 <= len b1
   holds (b1 *(b2,b4)) `1 < (b1 *(b3,b5)) `1;

:: JORDAN1J:th 2
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 <= len b1 & 1 <= b3 & b3 <= len b1 & 1 <= b4 & b4 < b5 & b5 <= width b1
   holds (b1 *(b2,b4)) `2 < (b1 *(b3,b5)) `2;

:: JORDAN1J:funcreg 1
registration
  let a1 be Relation-like Function-like non empty FinSequence-like set;
  let a2 be Relation-like Function-like FinSequence-like set;
  cluster a1 ^' a2 -> Relation-like Function-like non empty FinSequence-like;
end;

:: JORDAN1J:th 3
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
   (L~ ((Cage(b1,b2)) -: E-max L~ Cage(b1,b2))) /\ L~ ((Cage(b1,b2)) :- E-max L~ Cage(b1,b2)) = {N-min L~ Cage(b1,b2),E-max L~ Cage(b1,b2)};

:: JORDAN1J: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
   Upper_Seq(b2,b1) = (Rotate(Cage(b2,b1),E-max L~ Cage(b2,b1))) :- W-min L~ Cage(b2,b1);

:: JORDAN1J:th 5
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   W-min L~ Cage(b2,b1) in proj2 Upper_Seq(b2,b1) &
    W-min L~ Cage(b2,b1) in L~ Upper_Seq(b2,b1);

:: JORDAN1J:th 6
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
   W-max L~ Cage(b2,b1) in proj2 Upper_Seq(b2,b1) &
    W-max L~ Cage(b2,b1) in L~ Upper_Seq(b2,b1);

:: JORDAN1J:th 7
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
   N-min L~ Cage(b2,b1) in proj2 Upper_Seq(b2,b1) &
    N-min L~ Cage(b2,b1) in L~ Upper_Seq(b2,b1);

:: JORDAN1J:th 8
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
   N-max L~ Cage(b2,b1) in proj2 Upper_Seq(b2,b1) &
    N-max L~ Cage(b2,b1) in L~ Upper_Seq(b2,b1);

:: JORDAN1J:th 9
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   E-max L~ Cage(b2,b1) in proj2 Upper_Seq(b2,b1) &
    E-max L~ Cage(b2,b1) in L~ Upper_Seq(b2,b1);

:: JORDAN1J:th 10
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   E-max L~ Cage(b2,b1) in proj2 Lower_Seq(b2,b1) &
    E-max L~ Cage(b2,b1) in L~ Lower_Seq(b2,b1);

:: JORDAN1J:th 11
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   E-min L~ Cage(b2,b1) in proj2 Lower_Seq(b2,b1) &
    E-min L~ Cage(b2,b1) in L~ Lower_Seq(b2,b1);

:: JORDAN1J:th 12
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   S-max L~ Cage(b2,b1) in proj2 Lower_Seq(b2,b1) &
    S-max L~ Cage(b2,b1) in L~ Lower_Seq(b2,b1);

:: JORDAN1J:th 13
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   S-min L~ Cage(b2,b1) in proj2 Lower_Seq(b2,b1) &
    S-min L~ Cage(b2,b1) in L~ Lower_Seq(b2,b1);

:: JORDAN1J:th 14
theorem
for b1 being Element of NAT
for b2 being compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   W-min L~ Cage(b2,b1) in proj2 Lower_Seq(b2,b1) &
    W-min L~ Cage(b2,b1) in L~ Lower_Seq(b2,b1);

:: JORDAN1J:th 15
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & N-min b2 in b1
   holds N-min b1 = N-min b2;

:: JORDAN1J:th 16
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & N-max b2 in b1
   holds N-max b1 = N-max b2;

:: JORDAN1J:th 17
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & E-min b2 in b1
   holds E-min b1 = E-min b2;

:: JORDAN1J:th 18
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & E-max b2 in b1
   holds E-max b1 = E-max b2;

:: JORDAN1J:th 19
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & S-min b2 in b1
   holds S-min b1 = S-min b2;

:: JORDAN1J:th 20
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & S-max b2 in b1
   holds S-max b1 = S-max b2;

:: JORDAN1J:th 21
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & W-min b2 in b1
   holds W-min b1 = W-min b2;

:: JORDAN1J:th 22
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & W-max b2 in b1
   holds W-max b1 = W-max b2;

:: JORDAN1J:th 23
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st N-bound b1 <= N-bound b2
   holds N-bound (b1 \/ b2) = N-bound b2;

:: JORDAN1J:th 24
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st E-bound b1 <= E-bound b2
   holds E-bound (b1 \/ b2) = E-bound b2;

:: JORDAN1J:th 25
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st S-bound b1 <= S-bound b2
   holds S-bound (b1 \/ b2) = S-bound b1;

:: JORDAN1J:th 26
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st W-bound b1 <= W-bound b2
   holds W-bound (b1 \/ b2) = W-bound b1;

:: JORDAN1J:th 27
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st N-bound b1 < N-bound b2
   holds N-min (b1 \/ b2) = N-min b2;

:: JORDAN1J:th 28
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st N-bound b1 < N-bound b2
   holds N-max (b1 \/ b2) = N-max b2;

:: JORDAN1J:th 29
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st E-bound b1 < E-bound b2
   holds E-min (b1 \/ b2) = E-min b2;

:: JORDAN1J:th 30
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st E-bound b1 < E-bound b2
   holds E-max (b1 \/ b2) = E-max b2;

:: JORDAN1J:th 31
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st S-bound b1 < S-bound b2
   holds S-min (b1 \/ b2) = S-min b1;

:: JORDAN1J:th 32
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st S-bound b1 < S-bound b2
   holds S-max (b1 \/ b2) = S-max b1;

:: JORDAN1J:th 33
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st W-bound b1 < W-bound b2
   holds W-min (b1 \/ b2) = W-min b1;

:: JORDAN1J:th 34
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st W-bound b1 < W-bound b2
   holds W-max (b1 \/ b2) = W-max b1;

:: JORDAN1J:th 35
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;

:: JORDAN1J:th 36
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 the carrier of TOP-REAL 2
for b4 being connected Element of bool the carrier of TOP-REAL 2
      st b2 in RightComp b1 & b3 in LeftComp b1 & b2 in b4 & b3 in b4
   holds b4 meets L~ b1;

:: JORDAN1J:exreg 1
registration
  cluster Relation-like Function-like one-to-one non constant non empty finite FinSequence-like non trivial special unfolded s.n.c. being_S-Seq s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
end;

:: JORDAN1J:th 37
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1
   holds L_Cut(b1,b2) = mid(b1,b2 .. b1,len b1);

:: JORDAN1J:th 38
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 being_S-Seq FinSequence of the carrier of TOP-REAL 2
   st b2 is_sequence_on b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in proj2 b2
   holds R_Cut(b2,b3) is_sequence_on b1;

:: JORDAN1J:th 39
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 being_S-Seq FinSequence of the carrier of TOP-REAL 2
   st b2 is_sequence_on b1
for b3 being Element of the carrier of TOP-REAL 2
      st b3 in proj2 b2
   holds L_Cut(b2,b3) is_sequence_on b1;

:: JORDAN1J:th 40
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 FinSequence of the carrier of TOP-REAL 2
   st b2 is_sequence_on b1
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 <= len b1 & 1 <= b4 & b4 <= width b1 & b1 *(b3,b4) in L~ b2
   holds b1 *(b3,b4) in proj2 b2;

:: JORDAN1J:th 41
theorem
for b1 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b2 being FinSequence of the carrier of TOP-REAL 2
      st b2 is unfolded &
         b2 is s.n.c. &
         b2 is one-to-one &
         (L~ b1) /\ L~ b2 = {b1 /. 1} &
         b1 /. 1 = b2 /. len b2 &
         (for b3 being Element of NAT
               st 1 <= b3 & b3 + 2 <= len b1
            holds (LSeg(b1,b3)) /\ LSeg(b1 /. len b1,b2 /. 1) = {}) &
         (for b3 being Element of NAT
               st 2 <= b3 & b3 + 1 <= len b2
            holds (LSeg(b2,b3)) /\ LSeg(b1 /. len b1,b2 /. 1) = {})
   holds b1 ^ b2 is s.c.c.;

:: JORDAN1J:th 42
theorem
for b1 being Element of NAT
for b2 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2 holds
   ex b3 being Element of NAT st
      1 <= b3 & b3 + 1 <= len Gauge(b2,b1) & W-min b2 in cell(Gauge(b2,b1),1,b3) & W-min b2 <> (Gauge(b2,b1)) *(2,b3);

:: JORDAN1J:th 43
theorem
for b1 being being_S-Seq 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 . len b1 in L~ R_Cut(b1,b2)
   holds b1 . len b1 = b2;

:: JORDAN1J:th 44
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
   R_Cut(b1,b2) <> {};

:: JORDAN1J:th 45
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 (R_Cut(b1,b2)) /. len R_Cut(b1,b2) = b2;

:: JORDAN1J:th 46
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 the carrier of TOP-REAL 2
      st b3 in L~ Upper_Seq(b2,b1) & b3 `1 = E-bound L~ Cage(b2,b1)
   holds b3 = E-max L~ Cage(b2,b1);

:: JORDAN1J:th 47
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 the carrier of TOP-REAL 2
      st b3 in L~ Lower_Seq(b2,b1) & b3 `1 = W-bound L~ Cage(b2,b1)
   holds b3 = W-min L~ Cage(b2,b1);

:: JORDAN1J:th 48
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 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st 1 <= b4 & b4 < len b2 & b2 ^ b3 is_sequence_on b1
   holds left_cell(b2 ^ b3,b4,b1) = left_cell(b2,b4,b1) & right_cell(b2 ^ b3,b4,b1) = right_cell(b2,b4,b1);

:: JORDAN1J:th 49
theorem
for b1 being set
for b2, b3 being FinSequence of b1
for b4 being Element of NAT
      st b4 <= len b2
   holds (b2 ^' b3) | b4 = b2 | b4;

:: JORDAN1J:th 50
theorem
for b1 being set
for b2, b3 being FinSequence of b1 holds
(b2 ^' b3) | len b2 = b2;

:: JORDAN1J:th 51
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 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st 1 <= b4 & b4 < len b2 & b2 ^' b3 is_sequence_on b1
   holds left_cell(b2 ^' b3,b4,b1) = left_cell(b2,b4,b1) & right_cell(b2 ^' b3,b4,b1) = right_cell(b2,b4,b1);

:: JORDAN1J:th 52
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 being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st 1 <= b4 & b4 < b3 .. b2 & b2 is_sequence_on b1 & b3 in proj2 b2
   holds left_cell(R_Cut(b2,b3),b4,b1) = left_cell(b2,b4,b1) & right_cell(R_Cut(b2,b3),b4,b1) = right_cell(b2,b4,b1);

:: JORDAN1J:th 53
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 FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of NAT
      st 1 <= b4 & b4 < b3 .. b2 & b2 is_sequence_on b1
   holds left_cell(b2 -: b3,b4,b1) = left_cell(b2,b4,b1) & right_cell(b2 -: b3,b4,b1) = right_cell(b2,b4,b1);

:: JORDAN1J:th 54
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is unfolded &
         b1 is s.n.c. &
         b1 is one-to-one &
         b2 is unfolded &
         b2 is s.n.c. &
         b2 is one-to-one &
         b1 /. len b1 = b2 /. 1 &
         (L~ b1) /\ L~ b2 = {b2 /. 1}
   holds b1 ^' b2 is s.n.c.;

:: JORDAN1J:th 55
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 is one-to-one &
         b2 is one-to-one &
         (proj2 b1) /\ proj2 b2 c= {b2 /. 1}
   holds b1 ^' b2 is one-to-one;

:: JORDAN1J:th 56
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 proj2 b1 & b2 <> b1 . 1
   holds (Index(b2,b1)) + 1 = b2 .. b1;

:: JORDAN1J:th 57
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 &
         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 b4 <> b5;

:: JORDAN1J:th 58
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) &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN1J:th 59
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) &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Upper_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ L~ Lower_Seq(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN1J:th 60
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 &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Upper_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Lower_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Lower_Arc b2;

:: JORDAN1J:th 61
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 &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Upper_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b5)} &
         (LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5))) /\ Lower_Arc L~ Cage(b2,b1) = {(Gauge(b2,b1)) *(b3,b4)}
   holds LSeg((Gauge(b2,b1)) *(b3,b4),(Gauge(b2,b1)) *(b3,b5)) meets Upper_Arc b2;

:: JORDAN1J:th 62
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 (Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3) in Upper_Arc L~ Cage(b2,b1 + 1) &
         1 <= b3 &
         b3 <= width Gauge(b2,b1 + 1)
   holds LSeg((Gauge(b2,1)) *(Center Gauge(b2,1),1),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3)) meets Lower_Arc L~ Cage(b2,b1 + 1);

:: JORDAN1J:th 63
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) &
         (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4))) /\ Upper_Arc L~ Cage(b2,b1 + 1) = {(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4)} &
         (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4))) /\ Lower_Arc L~ Cage(b2,b1 + 1) = {(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3)}
   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;

:: JORDAN1J:th 64
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) &
         (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4))) /\ Upper_Arc L~ Cage(b2,b1 + 1) = {(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4)} &
         (LSeg((Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3),(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b4))) /\ Lower_Arc L~ Cage(b2,b1 + 1) = {(Gauge(b2,b1 + 1)) *(Center Gauge(b2,b1 + 1),b3)}
   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;

:: JORDAN1J:th 65
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & (W-min b2 in b1 or W-max b2 in b1)
   holds W-bound b1 = W-bound b2;

:: JORDAN1J:th 66
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & (E-min b2 in b1 or E-max b2 in b1)
   holds E-bound b1 = E-bound b2;

:: JORDAN1J:th 67
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & (N-min b2 in b1 or N-max b2 in b1)
   holds N-bound b1 = N-bound b2;

:: JORDAN1J:th 68
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
      st b1 c= b2 & (S-min b2 in b1 or S-max b2 in b1)
   holds S-bound b1 = S-bound b2;