Article JORDAN20, MML version 4.99.1005
:: JORDAN20:th 1
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 Segment(b1,b2,b3,b4,b4) = {b4};
:: JORDAN20:th 2
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of REAL
st b3 in LSeg(b1,b2) & b1 `1 <= b4 & b2 `1 <= b4
holds b3 `1 <= b4;
:: JORDAN20:th 3
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of REAL
st b3 in LSeg(b1,b2) & b4 <= b1 `1 & b4 <= b2 `1
holds b4 <= b3 `1;
:: JORDAN20:th 4
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of REAL
st b3 in LSeg(b1,b2) & b1 `1 < b4 & b2 `1 < b4
holds b3 `1 < b4;
:: JORDAN20:th 5
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of REAL
st b3 in LSeg(b1,b2) & b4 < b1 `1 & b4 < b2 `1
holds b4 < b3 `1;
:: JORDAN20:th 6
theorem
for b1 being Element of NAT
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st 1 <= b1 &
b1 < len b2 &
b3 in LSeg(b2,b1) &
b4 in LSeg(b2,b1) &
(b2 /. b1) `2 = (b2 /. (b1 + 1)) `2 &
(b2 /. (b1 + 1)) `1 < (b2 /. b1) `1 &
LE b3,b4,L~ b2,b2 /. 1,b2 /. len b2
holds b4 `1 <= b3 `1;
:: JORDAN20:th 7
theorem
for b1 being Element of NAT
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st 1 <= b1 &
b1 < len b2 &
b3 in LSeg(b2,b1) &
b4 in LSeg(b2,b1) &
(b2 /. b1) `2 = (b2 /. (b1 + 1)) `2 &
(b2 /. b1) `1 < (b2 /. (b1 + 1)) `1 &
LE b3,b4,L~ b2,b2 /. 1,b2 /. len b2
holds b3 `1 <= b4 `1;
:: JORDAN20:prednot 1 => JORDAN20:pred 1
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_Lin A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
b1 `1 < a5 &
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds b2 `1 <= a5));
end;
:: JORDAN20:dfs 1
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_Lin a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
b1 `1 < a5 &
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds b2 `1 <= a5));
:: JORDAN20:def 1
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
for b5 being Element of REAL holds
b4 is_Lin b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
b6 `1 < b5 &
LE b6,b4,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b6,b7,b1,b2,b3 & LE b7,b4,b1,b2,b3
holds b7 `1 <= b5));
:: JORDAN20:prednot 2 => JORDAN20:pred 2
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_Rin A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
a5 < b1 `1 &
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds a5 <= b2 `1));
end;
:: JORDAN20:dfs 2
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_Rin a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
a5 < b1 `1 &
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds a5 <= b2 `1));
:: JORDAN20:def 2
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
for b5 being Element of REAL holds
b4 is_Rin b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
b5 < b6 `1 &
LE b6,b4,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b6,b7,b1,b2,b3 & LE b7,b4,b1,b2,b3
holds b5 <= b7 `1));
:: JORDAN20:prednot 3 => JORDAN20:pred 3
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_Lout A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
b1 `1 < a5 &
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds b2 `1 <= a5));
end;
:: JORDAN20:dfs 3
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_Lout a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
b1 `1 < a5 &
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds b2 `1 <= a5));
:: JORDAN20:def 3
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
for b5 being Element of REAL holds
b4 is_Lout b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
b6 `1 < b5 &
LE b4,b6,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b7,b6,b1,b2,b3 & LE b4,b7,b1,b2,b3
holds b7 `1 <= b5));
:: JORDAN20:prednot 4 => JORDAN20:pred 4
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_Rout A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
a5 < b1 `1 &
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds a5 <= b2 `1));
end;
:: JORDAN20:dfs 4
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_Rout a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
a5 < b1 `1 &
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds a5 <= b2 `1));
:: JORDAN20:def 4
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
for b5 being Element of REAL holds
b4 is_Rout b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
b5 < b6 `1 &
LE b4,b6,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b7,b6,b1,b2,b3 & LE b4,b7,b1,b2,b3
holds b5 <= b7 `1));
:: JORDAN20:prednot 5 => JORDAN20:pred 5
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_OSin A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds b2 `1 = a5) &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & b2 <> b1
holds (ex b3 being Element of the carrier of TOP-REAL 2 st
LE b2,b3,a1,a2,a3 & LE b3,b1,a1,a2,a3 & a5 < b3 `1) &
(ex b3 being Element of the carrier of TOP-REAL 2 st
LE b2,b3,a1,a2,a3 & LE b3,b1,a1,a2,a3 & b3 `1 < a5)));
end;
:: JORDAN20:dfs 5
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_OSin a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
LE b1,a4,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & LE b2,a4,a1,a2,a3
holds b2 `1 = a5) &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & b2 <> b1
holds (ex b3 being Element of the carrier of TOP-REAL 2 st
LE b2,b3,a1,a2,a3 & LE b3,b1,a1,a2,a3 & a5 < b3 `1) &
(ex b3 being Element of the carrier of TOP-REAL 2 st
LE b2,b3,a1,a2,a3 & LE b3,b1,a1,a2,a3 & b3 `1 < a5)));
:: JORDAN20:def 5
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
for b5 being Element of REAL holds
b4 is_OSin b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
LE b6,b4,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b6,b7,b1,b2,b3 & LE b7,b4,b1,b2,b3
holds b7 `1 = b5) &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b7,b6,b1,b2,b3 & b7 <> b6
holds (ex b8 being Element of the carrier of TOP-REAL 2 st
LE b7,b8,b1,b2,b3 & LE b8,b6,b1,b2,b3 & b5 < b8 `1) &
(ex b8 being Element of the carrier of TOP-REAL 2 st
LE b7,b8,b1,b2,b3 & LE b8,b6,b1,b2,b3 & b8 `1 < b5)));
:: JORDAN20:prednot 6 => JORDAN20:pred 6
definition
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
pred A4 is_OSout A1,A2,A3,A5 means
a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds b2 `1 = a5) &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & b2 <> b1
holds (ex b3 being Element of the carrier of TOP-REAL 2 st
LE b3,b2,a1,a2,a3 & LE b1,b3,a1,a2,a3 & a5 < b3 `1) &
(ex b3 being Element of the carrier of TOP-REAL 2 st
LE b3,b2,a1,a2,a3 & LE b1,b3,a1,a2,a3 & b3 `1 < a5)));
end;
:: JORDAN20:dfs 6
definiens
let a1 be Element of bool the carrier of TOP-REAL 2;
let a2, a3, a4 be Element of the carrier of TOP-REAL 2;
let a5 be Element of REAL;
To prove
a4 is_OSout a1,a2,a3,a5
it is sufficient to prove
thus a1 is_an_arc_of a2,a3 &
a4 in a1 &
a4 `1 = a5 &
(ex b1 being Element of the carrier of TOP-REAL 2 st
LE a4,b1,a1,a2,a3 &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b2,b1,a1,a2,a3 & LE a4,b2,a1,a2,a3
holds b2 `1 = a5) &
(for b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,a1,a2,a3 & b2 <> b1
holds (ex b3 being Element of the carrier of TOP-REAL 2 st
LE b3,b2,a1,a2,a3 & LE b1,b3,a1,a2,a3 & a5 < b3 `1) &
(ex b3 being Element of the carrier of TOP-REAL 2 st
LE b3,b2,a1,a2,a3 & LE b1,b3,a1,a2,a3 & b3 `1 < a5)));
:: JORDAN20:def 6
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
for b5 being Element of REAL holds
b4 is_OSout b1,b2,b3,b5
iff
b1 is_an_arc_of b2,b3 &
b4 in b1 &
b4 `1 = b5 &
(ex b6 being Element of the carrier of TOP-REAL 2 st
LE b4,b6,b1,b2,b3 &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b7,b6,b1,b2,b3 & LE b4,b7,b1,b2,b3
holds b7 `1 = b5) &
(for b7 being Element of the carrier of TOP-REAL 2
st LE b6,b7,b1,b2,b3 & b7 <> b6
holds (ex b8 being Element of the carrier of TOP-REAL 2 st
LE b8,b7,b1,b2,b3 & LE b6,b8,b1,b2,b3 & b5 < b8 `1) &
(ex b8 being Element of the carrier of TOP-REAL 2 st
LE b8,b7,b1,b2,b3 & LE b6,b8,b1,b2,b3 & b8 `1 < b5)));
:: JORDAN20:th 8
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
for b5 being Element of REAL
st b1 is_an_arc_of b2,b3 & b2 `1 <= b5 & b5 <= b3 `1
holds ex b6 being Element of the carrier of TOP-REAL 2 st
b6 in b1 & b6 `1 = b5;
:: JORDAN20:th 9
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of REAL
st b1 is_an_arc_of b2,b3 & b2 `1 < b5 & b5 < b3 `1 & b4 in b1 & b4 `1 = b5 & not b4 is_Lin b1,b2,b3,b5 & not b4 is_Rin b1,b2,b3,b5
holds b4 is_OSin b1,b2,b3,b5;
:: JORDAN20:th 10
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of REAL
st b1 is_an_arc_of b2,b3 & b2 `1 < b5 & b5 < b3 `1 & b4 in b1 & b4 `1 = b5 & not b4 is_Lout b1,b2,b3,b5 & not b4 is_Rout b1,b2,b3,b5
holds b4 is_OSout b1,b2,b3,b5;
:: JORDAN20:th 11
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of REAL
st b1 = [.0,b2.[
holds b1 is open(I[01]);
:: JORDAN20:th 12
theorem
for b1 being Element of bool the carrier of I[01]
for b2 being Element of REAL
st b1 = ].b2,1.]
holds b1 is open(I[01]);
:: JORDAN20:th 13
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3 being Element of bool the carrier of I[01]
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b5 being Element of REAL
st b5 <= 1 &
b2 = {b6 where b6 is Element of the carrier of TOP-REAL 2: ex b7 being Element of REAL st
0 <= b7 & b7 < b5 & b6 = b4 . b7} &
b3 = [.0,b5.[
holds b4 .: b3 = b2;
:: JORDAN20:th 14
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3 being Element of bool the carrier of I[01]
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b5 being Element of REAL
st 0 <= b5 &
b2 = {b6 where b6 is Element of the carrier of TOP-REAL 2: ex b7 being Element of REAL st
b5 < b7 & b7 <= 1 & b6 = b4 . b7} &
b3 = ].b5,1.]
holds b4 .: b3 = b2;
:: JORDAN20:th 15
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b4 being Element of REAL
st b4 <= 1 &
b3 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: ex b6 being Element of REAL st
0 <= b6 & b6 < b4 & b5 = b3 . b6}
holds b2 is open((TOP-REAL 2) | b1);
:: JORDAN20:th 16
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1
for b4 being Element of REAL
st 0 <= b4 &
b3 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b2 = {b5 where b5 is Element of the carrier of TOP-REAL 2: ex b6 being Element of REAL st
b4 < b6 & b6 <= 1 & b5 = b3 . b6}
holds b2 is open((TOP-REAL 2) | b1);
:: JORDAN20:th 17
theorem
for b1 being non empty TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
st b2 /\ b3 = {} & b2 \/ b3 = the carrier of b1 & b4 in b2 & b5 in b3 & b2 is open(b1) & b3 is open(b1)
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1
st b6 is continuous(I[01], b1) & b6 . 0 = b4
holds b6 . 1 <> b5;
:: JORDAN20:th 18
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | b1
for b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b3,b4 & b5 in b1 & b5 <> b3 & b5 <> b4 & b2 = b1 \ {b5}
holds b2 is connected(not TOP-REAL 2 | b1) &
(for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of ((TOP-REAL 2) | b1) | b2
st b6 is continuous(I[01], ((TOP-REAL 2) | b1) | b2) &
b6 . 0 = b3
holds b6 . 1 <> b4);
:: JORDAN20:th 19
theorem
for b1 being non empty 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 & not LE b4,b5,b1,b2,b3
holds LE b5,b4,b1,b2,b3;
:: JORDAN20:th 21
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being non empty Element of bool the carrier of TOP-REAL b1
st b4 is_an_arc_of b2,b3 & b5 is_an_arc_of b2,b3 & b5 c= b4
holds b5 = b4;
:: JORDAN20:th 22
theorem
for b1 being non empty 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 & b3 <> b4
holds Segment(b1,b2,b3,b4,b3) is_an_arc_of b4,b3;
:: JORDAN20:th 23
theorem
for b1 being non empty 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 b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & LE b5,b6,b1,b2,b3
holds (Segment(b1,b2,b3,b4,b5)) \/ Segment(b1,b2,b3,b5,b6) = Segment(b1,b2,b3,b4,b6);
:: JORDAN20:th 24
theorem
for b1 being non empty 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 b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & LE b5,b6,b1,b2,b3
holds (Segment(b1,b2,b3,b4,b5)) /\ Segment(b1,b2,b3,b5,b6) = {b5};
:: JORDAN20:th 25
theorem
for b1 being non empty 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 Segment(b1,b2,b3,b2,b3) = b1;
:: JORDAN20:th 28
theorem
for b1, b2 being non empty Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b3,b4 & b2 is_an_arc_of b5,b6 & LE b5,b6,b1,b3,b4 & b2 c= b1
holds b2 = Segment(b1,b3,b4,b5,b6);
:: JORDAN20:th 29
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Element of REAL
st b2 `1 < b7 & b7 < b3 `1 & b4 is_Lin b1,b2,b3,b7 & b5 `1 = b7 & LSeg(b4,b5) c= b1 & b6 in LSeg(b4,b5)
holds b6 is_Lin b1,b2,b3,b7;
:: JORDAN20:th 30
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Element of REAL
st b2 `1 < b7 & b7 < b3 `1 & b4 is_Rin b1,b2,b3,b7 & b5 `1 = b7 & LSeg(b4,b5) c= b1 & b6 in LSeg(b4,b5)
holds b6 is_Rin b1,b2,b3,b7;
:: JORDAN20:th 31
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Element of REAL
st b2 `1 < b7 & b7 < b3 `1 & b4 is_Lout b1,b2,b3,b7 & b5 `1 = b7 & LSeg(b4,b5) c= b1 & b6 in LSeg(b4,b5)
holds b6 is_Lout b1,b2,b3,b7;
:: JORDAN20:th 32
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Element of REAL
st b2 `1 < b7 & b7 < b3 `1 & b4 is_Rout b1,b2,b3,b7 & b5 `1 = b7 & LSeg(b4,b5) c= b1 & b6 in LSeg(b4,b5)
holds b6 is_Rout b1,b2,b3,b7;
:: JORDAN20:th 33
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of REAL
st b1 is_S-P_arc_joining b2,b3 & b2 `1 < b5 & b5 < b3 `1 & b4 in b1 & b4 `1 = b5 & not b4 is_Lin b1,b2,b3,b5
holds b4 is_Rin b1,b2,b3,b5;
:: JORDAN20:th 34
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of REAL
st b1 is_S-P_arc_joining b2,b3 & b2 `1 < b5 & b5 < b3 `1 & b4 in b1 & b4 `1 = b5 & not b4 is_Lout b1,b2,b3,b5
holds b4 is_Rout b1,b2,b3,b5;