Article JORDAN12, MML version 4.99.1005

:: JORDAN12:th 1
theorem
for b1 being Element of NAT
      st 1 < b1
   holds 0 < b1 -' 1;

:: JORDAN12:th 3
theorem
1 is odd;

:: JORDAN12:th 4
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
for b3 being Element of NAT
      st 1 <= b3 & b3 + 1 <= len b2
   holds b2 /. b3 in proj2 b2 & b2 /. (b3 + 1) in proj2 b2;

:: JORDAN12:condreg 1
registration
  cluster s.n.c. -> s.c.c. (FinSequence of the carrier of TOP-REAL 2);
end;

:: JORDAN12:th 5
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1 ^' b2 is unfolded & b1 ^' b2 is s.c.c. & 2 <= len b2
   holds b1 is unfolded & b1 is s.n.c.;

:: JORDAN12:th 6
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2 holds
L~ b1 c= L~ (b1 ^' b2);

:: JORDAN12:prednot 1 => JORDAN12:pred 1
definition
  let a1 be Element of NAT;
  let a2, a3 be FinSequence of the carrier of TOP-REAL a1;
  pred A2 is_in_general_position_wrt A3 means
    L~ a2 misses proj2 a3 &
     (for b1 being Element of NAT
           st 1 <= b1 & b1 < len a3
        holds (L~ a2) /\ LSeg(a3,b1) is trivial);
end;

:: JORDAN12:dfs 1
definiens
  let a1 be Element of NAT;
  let a2, a3 be FinSequence of the carrier of TOP-REAL a1;
To prove
     a2 is_in_general_position_wrt a3
it is sufficient to prove
  thus L~ a2 misses proj2 a3 &
     (for b1 being Element of NAT
           st 1 <= b1 & b1 < len a3
        holds (L~ a2) /\ LSeg(a3,b1) is trivial);

:: JORDAN12:def 1
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL b1 holds
   b2 is_in_general_position_wrt b3
iff
   L~ b2 misses proj2 b3 &
    (for b4 being Element of NAT
          st 1 <= b4 & b4 < len b3
       holds (L~ b2) /\ LSeg(b3,b4) is trivial);

:: JORDAN12:prednot 2 => JORDAN12:pred 2
definition
  let a1 be Element of NAT;
  let a2, a3 be FinSequence of the carrier of TOP-REAL a1;
  pred A2,A3 are_in_general_position means
    a2 is_in_general_position_wrt a3 & a3 is_in_general_position_wrt a2;
  symmetry;
::  for a1 being Element of NAT
::  for a2, a3 being FinSequence of the carrier of TOP-REAL a1
::        st a2,a3 are_in_general_position
::     holds a3,a2 are_in_general_position;
end;

:: JORDAN12:dfs 2
definiens
  let a1 be Element of NAT;
  let a2, a3 be FinSequence of the carrier of TOP-REAL a1;
To prove
     a2,a3 are_in_general_position
it is sufficient to prove
  thus a2 is_in_general_position_wrt a3 & a3 is_in_general_position_wrt a2;

:: JORDAN12:def 2
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL b1 holds
   b2,b3 are_in_general_position
iff
   b2 is_in_general_position_wrt b3 & b3 is_in_general_position_wrt b2;

:: JORDAN12:th 7
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
   st b2,b3 are_in_general_position
for b4 being FinSequence of the carrier of TOP-REAL 2
      st b4 = b3 | Seg b1
   holds b2,b4 are_in_general_position;

:: JORDAN12:th 8
theorem
for b1, b2, b3, b4 being FinSequence of the carrier of TOP-REAL 2
      st b1 ^' b2,b3 ^' b4 are_in_general_position
   holds b1 ^' b2,b3 are_in_general_position;

:: JORDAN12:th 9
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
      st 1 <= b1 & b1 + 1 <= len b3 & b2,b3 are_in_general_position
   holds b3 . b1 in (L~ b2) ` & b3 . (b1 + 1) in (L~ b2) `;

:: JORDAN12:th 10
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
   st b1,b2 are_in_general_position
for b3, b4 being Element of NAT
      st 1 <= b3 & b3 + 1 <= len b1 & 1 <= b4 & b4 + 1 <= len b2
   holds (LSeg(b1,b3)) /\ LSeg(b2,b4) is trivial;

:: JORDAN12:th 11
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2 holds
INTERSECTION({LSeg(b1,b3) where b3 is Element of NAT: 1 <= b3 & b3 + 1 <= len b1},{LSeg(b2,b3) where b3 is Element of NAT: 1 <= b3 & b3 + 1 <= len b2}) is finite;

:: JORDAN12:th 12
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
      st b1,b2 are_in_general_position
   holds (L~ b1) /\ L~ b2 is finite;

:: JORDAN12:th 13
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
   st b1,b2 are_in_general_position
for b3 being Element of NAT holds
   (L~ b1) /\ LSeg(b2,b3) is finite;

:: JORDAN12:th 14
theorem
for b1 being non empty non constant 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
      st LSeg(b2,b3) misses L~ b1
   holds ex b4 being Element of bool the carrier of TOP-REAL 2 st
      b4 is_a_component_of (L~ b1) ` & b2 in b4 & b3 in b4;

:: JORDAN12:th 15
theorem
for b1, b2 being set
for b3 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
      ex b4 being Element of bool the carrier of TOP-REAL 2 st
         b4 is_a_component_of (L~ b3) ` & b1 in b4 & b2 in b4
   iff
      (b1 in RightComp b3 & b2 in RightComp b3 or b1 in LeftComp b3 & b2 in LeftComp b3);

:: JORDAN12:th 16
theorem
for b1, b2 being set
for b3 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
      b1 in (L~ b3) ` &
       b2 in (L~ b3) ` &
       (for b4 being Element of bool the carrier of TOP-REAL 2
             st b4 is_a_component_of (L~ b3) ` & b1 in b4
          holds not b2 in b4)
   iff
      (b1 in LeftComp b3 & b2 in RightComp b3 or b1 in RightComp b3 & b2 in LeftComp b3);

:: JORDAN12:th 17
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being set
      st (ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_a_component_of (L~ b1) ` & b2 in b5 & b3 in b5) &
         (ex b5 being Element of bool the carrier of TOP-REAL 2 st
            b5 is_a_component_of (L~ b1) ` & b3 in b5 & b4 in b5)
   holds ex b5 being Element of bool the carrier of TOP-REAL 2 st
      b5 is_a_component_of (L~ b1) ` & b2 in b5 & b4 in b5;

:: JORDAN12:th 18
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being set
      st b2 in (L~ b1) ` &
         b3 in (L~ b1) ` &
         b4 in (L~ b1) ` &
         (for b5 being Element of bool the carrier of TOP-REAL 2
               st b5 is_a_component_of (L~ b1) ` & b2 in b5
            holds not b3 in b5) &
         (for b5 being Element of bool the carrier of TOP-REAL 2
               st b5 is_a_component_of (L~ b1) ` & b3 in b5
            holds not b4 in b5)
   holds ex b5 being Element of bool the carrier of TOP-REAL 2 st
      b5 is_a_component_of (L~ b1) ` & b2 in b5 & b4 in b5;

:: JORDAN12:th 19
theorem
for b1 being Element of NAT
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 <= len b2
   holds v_strip(b2,b1) is convex(2);

:: JORDAN12:th 20
theorem
for b1 being Element of NAT
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 <= width b2
   holds h_strip(b2,b1) is convex(2);

:: JORDAN12:th 21
theorem
for b1, b2 being Element of NAT
for b3 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 <= len b3 & b2 <= width b3
   holds cell(b3,b1,b2) is convex(2);

:: JORDAN12:th 22
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds left_cell(b1,b2) is convex(2);

:: JORDAN12:th 23
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
      st 1 <= b2 & b2 + 1 <= len b1
   holds left_cell(b1,b2,GoB b1) is convex(2) & right_cell(b1,b2,GoB b1) is convex(2);

:: JORDAN12:th 24
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b4 being Element of the carrier of TOP-REAL 2
      st b4 in LSeg(b1,b2) &
         (ex b5 being set st
            (L~ b3) /\ LSeg(b1,b2) = {b5}) &
         not b4 in L~ b3 &
         L~ b3 meets LSeg(b1,b4)
   holds L~ b3 misses LSeg(b4,b2);

:: JORDAN12:th 25
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st LSeg(b1,b2) is vertical & LSeg(b3,b4) is vertical & LSeg(b1,b2) meets LSeg(b3,b4)
   holds b1 `1 = b3 `1;

:: JORDAN12:th 26
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st not b1 in LSeg(b2,b3) & b2 `2 = b3 `2 & b3 `2 = b1 `2 & not b2 in LSeg(b1,b3)
   holds b3 in LSeg(b1,b2);

:: JORDAN12:th 27
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st not b1 in LSeg(b2,b3) & b2 `1 = b3 `1 & b3 `1 = b1 `1 & not b2 in LSeg(b1,b3)
   holds b3 in LSeg(b1,b2);

:: JORDAN12:th 28
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
      st b1 <> b2 & b1 <> b3 & b1 in LSeg(b2,b3)
   holds not b2 in LSeg(b1,b3);

:: JORDAN12:th 29
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
      st not b4 in LSeg(b2,b3) &
         b1 in LSeg(b2,b3) &
         b1 <> b2 &
         b1 <> b3 &
         (b2 `1 = b3 `1 & b3 `1 = b4 `1 or b2 `2 = b3 `2 & b3 `2 = b4 `2) &
         not b2 in LSeg(b4,b1)
   holds b3 in LSeg(b4,b1);

:: JORDAN12:th 30
theorem
for b1, b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
      st (b1 `1 = b2 `1 & b3 `1 = b4 `1 or b1 `2 = b2 `2 & b3 `2 = b4 `2) &
         (LSeg(b1,b2)) /\ LSeg(b3,b4) = {b5} &
         b5 <> b1 &
         b5 <> b2
   holds b5 = b3;

:: JORDAN12:th 31
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
   st (L~ b4) /\ LSeg(b2,b3) = {b1}
for b5 being Element of the carrier of TOP-REAL 2
      st not b5 in LSeg(b2,b3) &
         not b2 in L~ b4 &
         not b3 in L~ b4 &
         (b2 `1 = b3 `1 & b2 `1 = b5 `1 or b2 `2 = b3 `2 & b2 `2 = b5 `2) &
         (ex b6 being Element of NAT st
            1 <= b6 &
             b6 + 1 <= len b4 &
             (b5 in right_cell(b4,b6,GoB b4) or b5 in left_cell(b4,b6,GoB b4)) &
             b1 in LSeg(b4,b6)) &
         not b5 in L~ b4 &
         (for b6 being Element of bool the carrier of TOP-REAL 2
               st b6 is_a_component_of (L~ b4) ` & b5 in b6
            holds not b2 in b6)
   holds ex b6 being Element of bool the carrier of TOP-REAL 2 st
      b6 is_a_component_of (L~ b4) ` & b5 in b6 & b3 in b6;

:: JORDAN12:th 32
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
   st (L~ b1) /\ LSeg(b2,b3) = {b4}
for b5, b6 being Element of the carrier of TOP-REAL 2
   st not b2 in L~ b1 &
      not b3 in L~ b1 &
      (b2 `1 = b3 `1 & b2 `1 = b5 `1 & b5 `1 = b6 `1 or b2 `2 = b3 `2 & b2 `2 = b5 `2 & b5 `2 = b6 `2) &
      (ex b7 being Element of NAT st
         1 <= b7 & b7 + 1 <= len b1 & b5 in left_cell(b1,b7,GoB b1) & b6 in right_cell(b1,b7,GoB b1) & b4 in LSeg(b1,b7)) &
      not b5 in L~ b1 &
      not b6 in L~ b1
for b7 being Element of bool the carrier of TOP-REAL 2
      st b7 is_a_component_of (L~ b1) ` & b2 in b7
   holds not b3 in b7;

:: JORDAN12:th 33
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
   st (L~ b2) /\ LSeg(b3,b4) = {b1} &
      (b3 `1 = b4 `1 or b3 `2 = b4 `2) &
      not b3 in L~ b2 &
      not b4 in L~ b2 &
      proj2 b2 misses LSeg(b3,b4)
for b5 being Element of bool the carrier of TOP-REAL 2
      st b5 is_a_component_of (L~ b2) ` & b3 in b5
   holds not b4 in b5;

:: JORDAN12:th 34
theorem
for b1 being non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2 being special FinSequence of the carrier of TOP-REAL 2
   st b1,b2 are_in_general_position
for b3 being Element of NAT
      st 1 <= b3 & b3 + 1 <= len b2
   holds    Card ((L~ b1) /\ LSeg(b2,b3)) is even Element of NAT
   iff
      ex b4 being Element of bool the carrier of TOP-REAL 2 st
         b4 is_a_component_of (L~ b1) ` & b2 . b3 in b4 & b2 . (b3 + 1) in b4;

:: JORDAN12:th 35
theorem
for b1, b2, b3 being special FinSequence of the carrier of TOP-REAL 2
      st b1 ^' b2 is non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 &
         b1 ^' b2,b3 are_in_general_position &
         2 <= len b3 &
         b3 is unfolded &
         b3 is s.n.c.
   holds    Card ((L~ (b1 ^' b2)) /\ L~ b3) is even Element of NAT
   iff
      ex b4 being Element of bool the carrier of TOP-REAL 2 st
         b4 is_a_component_of (L~ (b1 ^' b2)) ` & b3 . 1 in b4 & b3 . len b3 in b4;

:: JORDAN12:th 36
theorem
for b1, b2, b3, b4 being special FinSequence of the carrier of TOP-REAL 2
   st b1 ^' b2 is non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 &
      b3 ^' b4 is non empty non constant circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 &
      L~ b1 misses L~ b4 &
      L~ b2 misses L~ b3 &
      b1 ^' b2,b3 ^' b4 are_in_general_position
for b5, b6, b7, b8 being Element of the carrier of TOP-REAL 2
      st b1 . 1 = b5 &
         b1 . len b1 = b6 &
         b3 . 1 = b7 &
         b3 . len b3 = b8 &
         b1 /. len b1 = b2 /. 1 &
         b3 /. len b3 = b4 /. 1 &
         b5 in (L~ b1) /\ L~ b2 &
         b7 in (L~ b3) /\ L~ b4 &
         (ex b9 being Element of bool the carrier of TOP-REAL 2 st
            b9 is_a_component_of (L~ (b1 ^' b2)) ` & b7 in b9 & b8 in b9)
   holds ex b9 being Element of bool the carrier of TOP-REAL 2 st
      b9 is_a_component_of (L~ (b3 ^' b4)) ` & b5 in b9 & b6 in b9;