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;