Article JORDAN9, MML version 4.99.1005
:: JORDAN9:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2 is connected(b1) & b3 is_a_component_of b5 & b4 is_a_component_of b5 & b2 meets b3 & b2 meets b4 & b2 c= b5
holds b3 = b4;
:: JORDAN9:th 4
theorem
for b1 being non empty set
for b2, b3 being FinSequence of b1
st for b4 being Element of NAT holds
b2 | b4 = b3 | b4
holds b2 = b3;
:: JORDAN9:th 5
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
st b1 in dom b3
holds ex b4 being Element of NAT st
b4 in dom Rev b3 & b1 + b4 = (len b3) + 1 & b3 /. b1 = (Rev b3) /. b4;
:: JORDAN9:th 6
theorem
for b1 being Element of NAT
for b2 being non empty set
for b3 being FinSequence of b2
st b1 in dom Rev b3
holds ex b4 being Element of NAT st
b4 in dom b3 & b1 + b4 = (len b3) + 1 & (Rev b3) /. b1 = b3 /. b4;
:: JORDAN9:th 7
theorem
for b1 being non empty set
for b2 being tabular FinSequence of b1 *
for b3 being FinSequence of b1 holds
b3 is_sequence_on b2
iff
Rev b3 is_sequence_on b2;
:: JORDAN9:th 8
theorem
for b1 being Element of NAT
for b2 being tabular FinSequence of (the carrier of TOP-REAL 2) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & 1 <= b1 & b1 <= len b3
holds b3 /. b1 in Values b2;
:: JORDAN9:th 9
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being set
st b1 <= len b2 & b3 in L~ (b2 /^ b1)
holds ex b4 being Element of NAT st
b1 + 1 <= b4 & b4 + 1 <= len b2 & b3 in LSeg(b2,b4);
:: JORDAN9:th 10
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3
holds b3 /. b1 in left_cell(b3,b1,b2) & b3 /. b1 in right_cell(b3,b1,b2);
:: JORDAN9:th 11
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3
holds Int left_cell(b3,b1,b2) <> {} & Int right_cell(b3,b1,b2) <> {};
:: JORDAN9:th 12
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3
holds Int left_cell(b3,b1,b2) is connected(TOP-REAL 2) & Int right_cell(b3,b1,b2) is connected(TOP-REAL 2);
:: JORDAN9:th 13
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3
holds Cl Int left_cell(b3,b1,b2) = left_cell(b3,b1,b2) &
Cl Int right_cell(b3,b1,b2) = right_cell(b3,b1,b2);
:: JORDAN9:th 14
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & LSeg(b3,b1) is horizontal
holds ex b4 being Element of NAT st
1 <= b4 &
b4 <= width b2 &
(for b5 being Element of the carrier of TOP-REAL 2
st b5 in LSeg(b3,b1)
holds b5 `2 = (b2 *(1,b4)) `2);
:: JORDAN9:th 15
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & LSeg(b3,b1) is vertical
holds ex b4 being Element of NAT st
1 <= b4 &
b4 <= len b2 &
(for b5 being Element of the carrier of TOP-REAL 2
st b5 in LSeg(b3,b1)
holds b5 `1 = (b2 *(b4,1)) `1);
:: JORDAN9:th 16
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) *
for b4 being FinSequence of the carrier of TOP-REAL 2
st b4 is_sequence_on b3 & b4 is special & b1 <= len b3 & b2 <= width b3
holds Int cell(b3,b1,b2) misses L~ b4;
:: JORDAN9:th 17
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & b3 is special & 1 <= b1 & b1 + 1 <= len b3
holds Int left_cell(b3,b1,b2) misses L~ b3 & Int right_cell(b3,b1,b2) misses L~ b3;
:: JORDAN9:th 18
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
holds (b3 *(b1,b2)) `1 = (b3 *(b1,b2 + 1)) `1 &
(b3 *(b1,b2)) `2 = (b3 *(b1 + 1,b2)) `2 &
(b3 *(b1 + 1,b2 + 1)) `1 = (b3 *(b1 + 1,b2)) `1 &
(b3 *(b1 + 1,b2 + 1)) `2 = (b3 *(b1,b2 + 1)) `2;
:: JORDAN9:th 19
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 the carrier of TOP-REAL 2
for b3, b4 being Element of NAT
st 1 <= b3 & b3 + 1 <= len b1 & 1 <= b4 & b4 + 1 <= width b1
holds b2 in cell(b1,b3,b4)
iff
(b1 *(b3,b4)) `1 <= b2 `1 &
b2 `1 <= (b1 *(b3 + 1,b4)) `1 &
(b1 *(b3,b4)) `2 <= b2 `2 &
b2 `2 <= (b1 *(b3,b4 + 1)) `2;
:: JORDAN9:th 20
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
holds cell(b3,b1,b2) = {|[b4,b5]| where b4 is Element of REAL, b5 is Element of REAL: (b3 *(b1,b2)) `1 <= b4 & b4 <= (b3 *(b1 + 1,b2)) `1 & (b3 *(b1,b2)) `2 <= b5 & b5 <= (b3 *(b1,b2 + 1)) `2};
:: JORDAN9: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) *
for b4 being Element of the carrier of TOP-REAL 2
st 1 <= b1 &
b1 + 1 <= len b3 &
1 <= b2 &
b2 + 1 <= width b3 &
b4 in Values b3 &
b4 in cell(b3,b1,b2) &
b4 <> b3 *(b1,b2) &
b4 <> b3 *(b1,b2 + 1) &
b4 <> b3 *(b1 + 1,b2 + 1)
holds b4 = b3 *(b1 + 1,b2);
:: JORDAN9:th 22
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 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3
holds b3 *(b1,b2) in cell(b3,b1,b2) &
b3 *(b1,b2 + 1) in cell(b3,b1,b2) &
b3 *(b1 + 1,b2 + 1) in cell(b3,b1,b2) &
b3 *(b1 + 1,b2) in cell(b3,b1,b2);
:: JORDAN9:th 23
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) *
for b4 being Element of the carrier of TOP-REAL 2
st 1 <= b1 & b1 + 1 <= len b3 & 1 <= b2 & b2 + 1 <= width b3 & b4 in Values b3 & b4 in cell(b3,b1,b2)
holds b4 is_extremal_in cell(b3,b1,b2);
:: JORDAN9:th 24
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
st 2 <= len b2 & 2 <= width b2 & b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3
holds ex b4, b5 being Element of NAT st
1 <= b4 & b4 + 1 <= len b2 & 1 <= b5 & b5 + 1 <= width b2 & LSeg(b3,b1) c= cell(b2,b4,b5);
:: JORDAN9:th 25
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) *
for b3 being FinSequence of the carrier of TOP-REAL 2
for b4 being Element of the carrier of TOP-REAL 2
st 2 <= len b2 & 2 <= width b2 & b3 is_sequence_on b2 & 1 <= b1 & b1 + 1 <= len b3 & b4 in Values b2 & b4 in LSeg(b3,b1) & b4 <> b3 /. b1
holds b4 = b3 /. (b1 + 1);
:: JORDAN9:th 26
theorem
for b1, b2, b3 being Element of NAT
for b4 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,b2] in Indices b4 & 1 <= b3 & b3 <= width b4
holds (b4 *(b1,b2)) `1 <= (b4 *(len b4,b3)) `1;
:: JORDAN9:th 27
theorem
for b1, b2, b3 being Element of NAT
for b4 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,b2] in Indices b4 & 1 <= b3 & b3 <= len b4
holds (b4 *(b1,b2)) `2 <= (b4 *(b3,width b4)) `2;
:: JORDAN9:th 28
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) *
for b3, b4 being FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on b2 & b3 is special & L~ b4 c= L~ b3 & 1 <= b1 & b1 + 1 <= len b3
for b5 being Element of bool the carrier of TOP-REAL 2
st (b5 = (right_cell(b3,b1,b2)) \ L~ b4 or b5 = (left_cell(b3,b1,b2)) \ L~ b4)
holds b5 is connected(TOP-REAL 2);
:: JORDAN9:th 29
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 non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b2 is_sequence_on b1
for b3 being Element of NAT
st 1 <= b3 & b3 + 1 <= len b2
holds (right_cell(b2,b3,b1)) \ L~ b2 c= RightComp b2 &
(left_cell(b2,b3,b1)) \ L~ b2 c= LeftComp b2;
:: JORDAN9:th 30
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being natural set holds
ex b3 being Element of NAT st
1 <= b3 &
b3 + 1 <= len Gauge(b1,b2) &
N-min b1 in cell(Gauge(b1,b2),b3,(width Gauge(b1,b2)) -' 1) &
N-min b1 <> (Gauge(b1,b2)) *(b3,(width Gauge(b1,b2)) -' 1);
:: JORDAN9:th 31
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being natural set
st 1 <= b3 &
b3 + 1 <= len Gauge(b1,b2) &
N-min b1 in cell(Gauge(b1,b2),b3,(width Gauge(b1,b2)) -' 1) &
N-min b1 <> (Gauge(b1,b2)) *(b3,(width Gauge(b1,b2)) -' 1) &
1 <= b4 &
b4 + 1 <= len Gauge(b1,b2) &
N-min b1 in cell(Gauge(b1,b2),b4,(width Gauge(b1,b2)) -' 1) &
N-min b1 <> (Gauge(b1,b2)) *(b4,(width Gauge(b1,b2)) -' 1)
holds b3 = b4;
:: JORDAN9:th 32
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being natural set
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b3 is_sequence_on Gauge(b1,b2) &
(for b4 being Element of NAT
st 1 <= b4 & b4 + 1 <= len b3
holds left_cell(b3,b4,Gauge(b1,b2)) misses b1 & right_cell(b3,b4,Gauge(b1,b2)) meets b1) &
(ex b4 being Element of NAT st
1 <= b4 &
b4 + 1 <= len Gauge(b1,b2) &
b3 /. 1 = (Gauge(b1,b2)) *(b4,width Gauge(b1,b2)) &
b3 /. 2 = (Gauge(b1,b2)) *(b4 + 1,width Gauge(b1,b2)) &
N-min b1 in cell(Gauge(b1,b2),b4,(width Gauge(b1,b2)) -' 1) &
N-min b1 <> (Gauge(b1,b2)) *(b4,(width Gauge(b1,b2)) -' 1))
holds N-min L~ b3 = b3 /. 1;
:: JORDAN9:funcnot 1 => JORDAN9:func 1
definition
let a1 be non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
let a2 be natural set;
assume a1 is connected(TOP-REAL 2);
func Cage(A1,A2) -> non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 means
it is_sequence_on Gauge(a1,a2) &
(ex b1 being Element of NAT st
1 <= b1 &
b1 + 1 <= len Gauge(a1,a2) &
it /. 1 = (Gauge(a1,a2)) *(b1,width Gauge(a1,a2)) &
it /. 2 = (Gauge(a1,a2)) *(b1 + 1,width Gauge(a1,a2)) &
N-min a1 in cell(Gauge(a1,a2),b1,(width Gauge(a1,a2)) -' 1) &
N-min a1 <> (Gauge(a1,a2)) *(b1,(width Gauge(a1,a2)) -' 1)) &
(for b1 being Element of NAT
st 1 <= b1 & b1 + 2 <= len it
holds (front_left_cell(it,b1,Gauge(a1,a2)) misses a1 & front_right_cell(it,b1,Gauge(a1,a2)) misses a1 implies it turns_right b1,Gauge(a1,a2)) &
(front_left_cell(it,b1,Gauge(a1,a2)) misses a1 & front_right_cell(it,b1,Gauge(a1,a2)) meets a1 implies it goes_straight b1,Gauge(a1,a2)) &
(front_left_cell(it,b1,Gauge(a1,a2)) misses a1 or it turns_left b1,Gauge(a1,a2)));
end;
:: JORDAN9:def 1
theorem
for b1 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
for b2 being natural set
st b1 is connected(TOP-REAL 2)
for b3 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2 holds
b3 = Cage(b1,b2)
iff
b3 is_sequence_on Gauge(b1,b2) &
(ex b4 being Element of NAT st
1 <= b4 &
b4 + 1 <= len Gauge(b1,b2) &
b3 /. 1 = (Gauge(b1,b2)) *(b4,width Gauge(b1,b2)) &
b3 /. 2 = (Gauge(b1,b2)) *(b4 + 1,width Gauge(b1,b2)) &
N-min b1 in cell(Gauge(b1,b2),b4,(width Gauge(b1,b2)) -' 1) &
N-min b1 <> (Gauge(b1,b2)) *(b4,(width Gauge(b1,b2)) -' 1)) &
(for b4 being Element of NAT
st 1 <= b4 & b4 + 2 <= len b3
holds (front_left_cell(b3,b4,Gauge(b1,b2)) misses b1 & front_right_cell(b3,b4,Gauge(b1,b2)) misses b1 implies b3 turns_right b4,Gauge(b1,b2)) &
(front_left_cell(b3,b4,Gauge(b1,b2)) misses b1 & front_right_cell(b3,b4,Gauge(b1,b2)) meets b1 implies b3 goes_straight b4,Gauge(b1,b2)) &
(front_left_cell(b3,b4,Gauge(b1,b2)) misses b1 or b3 turns_left b4,Gauge(b1,b2)));
:: JORDAN9:th 33
theorem
for b1, b2 being Element of NAT
for b3 being non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2
st b3 is connected(TOP-REAL 2) & 1 <= b1 & b1 + 1 <= len Cage(b3,b2)
holds left_cell(Cage(b3,b2),b1,Gauge(b3,b2)) misses b3 &
right_cell(Cage(b3,b2),b1,Gauge(b3,b2)) meets b3;
:: JORDAN9:th 34
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
st b2 is connected(TOP-REAL 2)
holds N-min L~ Cage(b2,b1) = (Cage(b2,b1)) /. 1;