Article JORDAN5C, MML version 4.99.1005
:: JORDAN5C:th 1
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
st b5 in b1 &
b5 in b2 &
b6 . b7 = b5 &
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b6 . 0 = b3 &
b6 . 1 = b4 &
0 <= b7 &
b7 <= 1 &
(for b8 being Element of REAL
st 0 <= b8 & b8 < b7
holds not b6 . b8 in b2)
for b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b9 being Element of REAL
st b8 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b8 . 0 = b3 & b8 . 1 = b4 & b8 . b9 = b5 & 0 <= b9 & b9 <= 1
for b10 being Element of REAL
st 0 <= b10 & b10 < b9
holds not b8 . b10 in b2;
:: JORDAN5C:funcnot 1 => JORDAN5C:func 1
definition
let a1, a2 be Element of bool the carrier of TOP-REAL 2;
let a3, a4 be Element of the carrier of TOP-REAL 2;
assume a1 meets a2 & a1 /\ a2 is closed(TOP-REAL 2) & a1 is_an_arc_of a3,a4;
func First_Point(A1,A3,A4,A2) -> Element of the carrier of TOP-REAL 2 means
it in a1 /\ a2 &
(for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
for b2 being Element of REAL
st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a3 & b1 . 1 = a4 & b1 . b2 = it & 0 <= b2 & b2 <= 1
for b3 being Element of REAL
st 0 <= b3 & b3 < b2
holds not b1 . b3 in a2);
end;
:: JORDAN5C:def 1
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
for b5 being Element of the carrier of TOP-REAL 2 holds
b5 = First_Point(b1,b3,b4,b2)
iff
b5 in b1 /\ b2 &
(for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b3 & b6 . 1 = b4 & b6 . b7 = b5 & 0 <= b7 & b7 <= 1
for b8 being Element of REAL
st 0 <= b8 & b8 < b7
holds not b6 . b8 in b2);
:: JORDAN5C:th 2
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b3 in b1 & b1 is_an_arc_of b4,b5 & b2 = {b3}
holds First_Point(b1,b4,b5,b2) = b3;
:: JORDAN5C:th 3
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b3 in b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
holds First_Point(b1,b3,b4,b2) = b3;
:: JORDAN5C:th 4
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
st b5 in b1 &
b5 in b2 &
b6 . b7 = b5 &
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b6 . 0 = b3 &
b6 . 1 = b4 &
0 <= b7 &
b7 <= 1 &
(for b8 being Element of REAL
st b8 <= 1 & b7 < b8
holds not b6 . b8 in b2)
for b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b9 being Element of REAL
st b8 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b8 . 0 = b3 & b8 . 1 = b4 & b8 . b9 = b5 & 0 <= b9 & b9 <= 1
for b10 being Element of REAL
st b10 <= 1 & b9 < b10
holds not b8 . b10 in b2;
:: JORDAN5C:funcnot 2 => JORDAN5C:func 2
definition
let a1, a2 be Element of bool the carrier of TOP-REAL 2;
let a3, a4 be Element of the carrier of TOP-REAL 2;
assume a1 meets a2 & a1 /\ a2 is closed(TOP-REAL 2) & a1 is_an_arc_of a3,a4;
func Last_Point(A1,A3,A4,A2) -> Element of the carrier of TOP-REAL 2 means
it in a1 /\ a2 &
(for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
for b2 being Element of REAL
st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a3 & b1 . 1 = a4 & b1 . b2 = it & 0 <= b2 & b2 <= 1
for b3 being Element of REAL
st b3 <= 1 & b2 < b3
holds not b1 . b3 in a2);
end;
:: JORDAN5C:def 2
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
for b5 being Element of the carrier of TOP-REAL 2 holds
b5 = Last_Point(b1,b3,b4,b2)
iff
b5 in b1 /\ b2 &
(for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7 being Element of REAL
st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b3 & b6 . 1 = b4 & b6 . b7 = b5 & 0 <= b7 & b7 <= 1
for b8 being Element of REAL
st b8 <= 1 & b7 < b8
holds not b6 . b8 in b2);
:: JORDAN5C:th 5
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b3 in b1 & b1 is_an_arc_of b4,b5 & b2 = {b3}
holds Last_Point(b1,b4,b5,b2) = b3;
:: JORDAN5C:th 6
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b4 in b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
holds Last_Point(b1,b3,b4,b2) = b4;
:: JORDAN5C:th 7
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 c= b2 & b1 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
holds First_Point(b1,b3,b4,b2) = b3 & Last_Point(b1,b3,b4,b2) = b4;
:: JORDAN5C:prednot 1 => JORDAN5C:pred 1
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4, a5 be Element of the carrier of TOP-REAL 2;
pred LE A4,A5,A1,A2,A3 means
a4 in a1 &
a5 in a1 &
(for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
for b2, b3 being Element of REAL
st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a2 & b1 . 1 = a3 & b1 . b2 = a4 & 0 <= b2 & b2 <= 1 & b1 . b3 = a5 & 0 <= b3 & b3 <= 1
holds b2 <= b3);
end;
:: JORDAN5C:dfs 3
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4, a5 be Element of the carrier of TOP-REAL 2;
To prove
LE a4,a5,a1,a2,a3
it is sufficient to prove
thus a4 in a1 &
a5 in a1 &
(for b1 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | a1
for b2, b3 being Element of REAL
st b1 is being_homeomorphism(I[01], (TOP-REAL 2) | a1) & b1 . 0 = a2 & b1 . 1 = a3 & b1 . b2 = a4 & 0 <= b2 & b2 <= 1 & b1 . b3 = a5 & 0 <= b3 & b3 <= 1
holds b2 <= b3);
:: JORDAN5C:def 3
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2 holds
LE b4,b5,b1,b2,b3
iff
b4 in b1 &
b5 in b1 &
(for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7, b8 being Element of REAL
st b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & 0 <= b7 & b7 <= 1 & b6 . b8 = b5 & 0 <= b8 & b8 <= 1
holds b7 <= b8);
:: JORDAN5C:th 8
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b7, b8 being Element of REAL
st b1 is_an_arc_of b2,b3 & b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & 0 <= b7 & b7 <= 1 & b6 . b8 = b5 & 0 <= b8 & b8 <= 1 & b7 <= b8
holds LE b4,b5,b1,b2,b3;
:: JORDAN5C:th 9
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b4 in b1
holds LE b4,b4,b1,b2,b3;
:: JORDAN5C:th 10
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1
holds LE b2,b4,b1,b2,b3 & LE b4,b3,b1,b2,b3;
:: JORDAN5C:th 11
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3
holds LE b2,b3,b1,b2,b3;
:: JORDAN5C:th 12
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & LE b5,b4,b1,b2,b3
holds b4 = b5;
:: JORDAN5C:th 13
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
st LE b4,b5,b1,b2,b3 & LE b5,b6,b1,b2,b3
holds LE b4,b6,b1,b2,b3;
:: JORDAN5C:th 14
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1 & b5 in b1 & b4 <> b5 & (LE b4,b5,b1,b2,b3 implies LE b5,b4,b1,b2,b3)
holds LE b5,b4,b1,b2,b3 & not LE b4,b5,b1,b2,b3;
:: JORDAN5C:th 15
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & (L~ b1) /\ b2 is closed(TOP-REAL 2) & b3 in L~ b1 & b3 in b2
holds LE First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b3,L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 16
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & (L~ b1) /\ b2 is closed(TOP-REAL 2) & b3 in L~ b1 & b3 in b2
holds LE b3,Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 17
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b3 <> b4 & LE b1,b2,LSeg(b3,b4),b3,b4
holds LE b1,b2,b3,b4;
:: JORDAN5C:th 18
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b3,b4 & b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2)
holds First_Point(b1,b3,b4,b2) = Last_Point(b1,b4,b3,b2) & Last_Point(b1,b3,b4,b2) = First_Point(b1,b4,b3,b2);
:: JORDAN5C:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
st L~ b1 meets b2 &
b2 is closed(TOP-REAL 2) &
b1 is being_S-Seq &
1 <= b3 &
b3 + 1 <= len b1 &
First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b3)
holds First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) = First_Point(LSeg(b1,b3),b1 /. b3,b1 /. (b3 + 1),b2);
:: JORDAN5C:th 20
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of NAT
st L~ b1 meets b2 &
b2 is closed(TOP-REAL 2) &
b1 is being_S-Seq &
1 <= b3 &
b3 + 1 <= len b1 &
Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b3)
holds Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) = Last_Point(LSeg(b1,b3),b1 /. b3,b1 /. (b3 + 1),b2);
:: JORDAN5C:th 21
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 &
b2 + 1 <= len b1 &
b1 is being_S-Seq &
First_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) in LSeg(b1,b2)
holds First_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) = b1 /. b2;
:: JORDAN5C:th 22
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 &
b2 + 1 <= len b1 &
b1 is being_S-Seq &
Last_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) in LSeg(b1,b2)
holds Last_Point(L~ b1,b1 /. 1,b1 /. len b1,LSeg(b1,b2)) = b1 /. (b2 + 1);
:: JORDAN5C:th 23
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st b1 is being_S-Seq & 1 <= b2 & b2 + 1 <= len b1
holds LE b1 /. b2,b1 /. (b2 + 1),L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 24
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b1 is being_S-Seq & 1 <= b2 & b2 <= b3 & b3 <= len b1
holds LE b1 /. b2,b1 /. b3,L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 25
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 is being_S-Seq & 1 <= b3 & b3 + 1 <= len b1 & b2 in LSeg(b1,b3)
holds LE b1 /. b3,b2,L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 26
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 is being_S-Seq & 1 <= b3 & b3 + 1 <= len b1 & b2 in LSeg(b1,b3)
holds LE b2,b1 /. (b3 + 1),L~ b1,b1 /. 1,b1 /. len b1;
:: JORDAN5C:th 27
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4, b5 being Element of NAT
st L~ b1 meets b2 &
b1 is being_S-Seq &
b2 is closed(TOP-REAL 2) &
First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b4) &
1 <= b4 &
b4 + 1 <= len b1 &
b3 in LSeg(b1,b5) &
1 <= b5 &
b5 + 1 <= len b1 &
b3 in b2 &
First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) <> b3
holds b4 <= b5 &
(b4 = b5 implies LE First_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b3,b1 /. b4,b1 /. (b4 + 1));
:: JORDAN5C:th 28
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4, b5 being Element of NAT
st L~ b1 meets b2 &
b1 is being_S-Seq &
b2 is closed(TOP-REAL 2) &
Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) in LSeg(b1,b4) &
1 <= b4 &
b4 + 1 <= len b1 &
b3 in LSeg(b1,b5) &
1 <= b5 &
b5 + 1 <= len b1 &
b3 in b2 &
Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2) <> b3
holds b5 <= b4 &
(b4 = b5 implies LE b3,Last_Point(L~ b1,b1 /. 1,b1 /. len b1,b2),b1 /. b4,b1 /. (b4 + 1));
:: JORDAN5C:th 29
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of NAT
st b2 in LSeg(b1,b4) &
b3 in LSeg(b1,b4) &
b1 is being_S-Seq &
1 <= b4 &
b4 + 1 <= len b1 &
LE b2,b3,L~ b1,b1 /. 1,b1 /. len b1
holds LE b2,b3,LSeg(b1,b4),b1 /. b4,b1 /. (b4 + 1);
:: JORDAN5C:th 30
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1 & b3 in L~ b1 & b1 is being_S-Seq & b2 <> b3
holds LE b2,b3,L~ b1,b1 /. 1,b1 /. len b1
iff
for b4, b5 being Element of NAT
st b2 in LSeg(b1,b4) & b3 in LSeg(b1,b5) & 1 <= b4 & b4 + 1 <= len b1 & 1 <= b5 & b5 + 1 <= len b1
holds b4 <= b5 &
(b4 = b5 implies LE b2,b3,b1 /. b4,b1 /. (b4 + 1));