Article JGRAPH_6, MML version 4.99.1005
:: JGRAPH_6:th 9
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2
st b2 <= b3 &
b4 in LSeg(|[b1,b2]|,|[b1,b3]|)
holds b4 `1 = b1 & b2 <= b4 `2 & b4 `2 <= b3;
:: JGRAPH_6:th 10
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2
st b2 < b3 & b4 `1 = b1 & b2 <= b4 `2 & b4 `2 <= b3
holds b4 in LSeg(|[b1,b2]|,|[b1,b3]|);
:: JGRAPH_6:th 11
theorem
for b1, b2, b3 being real set
for b4 being Element of the carrier of TOP-REAL 2
st b1 <= b2 &
b4 in LSeg(|[b1,b3]|,|[b2,b3]|)
holds b4 `2 = b3 & b1 <= b4 `1 & b4 `1 <= b2;
:: JGRAPH_6:th 12
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of I[01]
st b3 = [.b1,b2.]
holds b3 is closed(I[01]);
:: JGRAPH_6:th 13
theorem
for b1 being TopStruct
for b2, b3 being non empty TopStruct
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3 holds
dom b4 = dom b5 & dom b4 = the carrier of b1 & dom b4 = [#] b1;
:: JGRAPH_6:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1 holds
ex b3 being Function-like quasi_total Relation of the carrier of b1 | b2,the carrier of b1 st
(for b4 being Element of the carrier of b1 | b2 holds
b3 . b4 = b4) &
b3 is continuous(b1 | b2, b1);
:: JGRAPH_6:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being real set
st b2 is continuous(b1, R^1)
holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
(for b5 being Element of the carrier of b1
for b6 being real set
st b2 . b5 = b6
holds b4 . b5 = b6 - b3) &
b4 is continuous(b1, R^1);
:: JGRAPH_6:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being real set
st b2 is continuous(b1, R^1)
holds ex b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
(for b5 being Element of the carrier of b1
for b6 being real set
st b2 . b5 = b6
holds b4 . b5 = b3 - b6) &
b4 is continuous(b1, R^1);
:: JGRAPH_6:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of NAT
for b3 being Element of the carrier of TOP-REAL b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b4 is continuous(b1, R^1)
holds ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of TOP-REAL b2 st
(for b6 being Element of the carrier of b1 holds
b5 . b6 = (b4 . b6) * b3) &
b5 is continuous(b1, TOP-REAL b2);
:: JGRAPH_6:th 18
theorem
Sq_Circ . |[- 1,0]| = |[- 1,0]|;
:: JGRAPH_6:th 19
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2
st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = 1}
holds Sq_Circ . |[- 1,0]| = W-min b1;
:: JGRAPH_6:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of NAT
for b3, b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of TOP-REAL b2
st b3 is continuous(b1, TOP-REAL b2) & b4 is continuous(b1, TOP-REAL b2)
holds ex b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of TOP-REAL b2 st
(for b6 being Element of the carrier of b1 holds
b5 . b6 = (b3 . b6) + (b4 . b6)) &
b5 is continuous(b1, TOP-REAL b2);
:: JGRAPH_6:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of NAT
for b3, b4 being Element of the carrier of TOP-REAL b2
for b5, b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b5 is continuous(b1, R^1) & b6 is continuous(b1, R^1)
holds ex b7 being Function-like quasi_total Relation of the carrier of b1,the carrier of TOP-REAL b2 st
(for b8 being Element of the carrier of b1 holds
b7 . b8 = ((b5 . b8) * b3) + ((b6 . b8) * b4)) &
b7 is continuous(b1, TOP-REAL b2);
:: JGRAPH_6:th 22
theorem
for b1 being Relation-like Function-like set
for b2 being set
st b1 is one-to-one & b2 c= proj1 b1
holds b1 " .: (b1 .: b2) = b2;
:: JGRAPH_6:th 23
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b3, b4, b5, b6, b7 being Element of bool the carrier of TOP-REAL 2
for b8, b9 being Element of the carrier of I[01]
st b8 = 0 &
b9 = 1 &
b1 is continuous(I[01], TOP-REAL 2) &
b1 is one-to-one &
b2 is continuous(I[01], TOP-REAL 2) &
b2 is one-to-one &
b3 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| <= 1} &
b4 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & - (b10 `1) <= b10 `2} &
b5 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & b10 `2 <= - (b10 `1)} &
b6 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & - (b10 `1) <= b10 `2} &
b7 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & b10 `2 <= - (b10 `1)} &
b1 . b8 in b4 &
b1 . b9 in b5 &
b2 . b8 in b6 &
b2 . b9 in b7 &
rng b1 c= b3 &
rng b2 c= b3
holds rng b1 meets rng b2;
:: JGRAPH_6:th 24
theorem
for b1, b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
for b3, b4, b5, b6, b7 being Element of bool the carrier of TOP-REAL 2
for b8, b9 being Element of the carrier of I[01]
st b8 = 0 &
b9 = 1 &
b1 is continuous(I[01], TOP-REAL 2) &
b1 is one-to-one &
b2 is continuous(I[01], TOP-REAL 2) &
b2 is one-to-one &
b3 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| <= 1} &
b4 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & - (b10 `1) <= b10 `2} &
b5 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & b10 `2 <= - (b10 `1)} &
b6 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `1 <= b10 `2 & - (b10 `1) <= b10 `2} &
b7 = {b10 where b10 is Element of the carrier of TOP-REAL 2: |.b10.| = 1 & b10 `2 <= b10 `1 & b10 `2 <= - (b10 `1)} &
b1 . b8 in b4 &
b1 . b9 in b5 &
b2 . b8 in b7 &
b2 . b9 in b6 &
rng b1 c= b3 &
rng b2 c= b3
holds rng b1 meets rng b2;
:: JGRAPH_6:th 25
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
LE b1,b2,b5 &
LE b2,b3,b5 &
LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
st b7 is continuous(I[01], TOP-REAL 2) &
b7 is one-to-one &
b8 is continuous(I[01], TOP-REAL 2) &
b8 is one-to-one &
b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: |.b9.| <= 1} &
b7 . 0 = b3 &
b7 . 1 = b1 &
b8 . 0 = b2 &
b8 . 1 = b4 &
rng b7 c= b6 &
rng b8 c= b6
holds rng b7 meets rng b8;
:: JGRAPH_6:th 26
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
LE b1,b2,b5 &
LE b2,b3,b5 &
LE b3,b4,b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
st b7 is continuous(I[01], TOP-REAL 2) &
b7 is one-to-one &
b8 is continuous(I[01], TOP-REAL 2) &
b8 is one-to-one &
b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: |.b9.| <= 1} &
b7 . 0 = b3 &
b7 . 1 = b1 &
b8 . 0 = b4 &
b8 . 1 = b2 &
rng b7 c= b6 &
rng b8 c= b6
holds rng b7 meets rng b8;
:: JGRAPH_6:th 27
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Element of bool the carrier of TOP-REAL 2
st b5 = {b7 where b7 is Element of the carrier of TOP-REAL 2: |.b7.| = 1} &
b1,b2,b3,b4 are_in_this_order_on b5
for b7, b8 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
st b7 is continuous(I[01], TOP-REAL 2) &
b7 is one-to-one &
b8 is continuous(I[01], TOP-REAL 2) &
b8 is one-to-one &
b6 = {b9 where b9 is Element of the carrier of TOP-REAL 2: |.b9.| <= 1} &
b7 . 0 = b1 &
b7 . 1 = b3 &
b8 . 0 = b2 &
b8 . 1 = b4 &
rng b7 c= b6 &
rng b8 c= b6
holds rng b7 meets rng b8;
:: JGRAPH_6:funcnot 1 => SPPOL_2:func 1
notation
let a1, a2, a3, a4 be real set;
synonym rectangle(a1,a2,a3,a4) for [.a1,a2,a3,a4.];
end;
:: JGRAPH_6:th 28
theorem
for b1, b2, b3, b4 being real set
for b5 being Element of the carrier of TOP-REAL 2
st b1 <= b2 & b3 <= b4 & b5 in [.b1,b2,b3,b4.]
holds b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 & b5 `2 <= b4;
:: JGRAPH_6:funcnot 2 => JGRAPH_6:func 1
definition
let a1, a2, a3, a4 be real set;
func inside_of_rectangle(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: a1 < b1 `1 & b1 `1 < a2 & a3 < b1 `2 & b1 `2 < a4};
end;
:: JGRAPH_6:def 1
theorem
for b1, b2, b3, b4 being real set holds
inside_of_rectangle(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 < b5 `1 & b5 `1 < b2 & b3 < b5 `2 & b5 `2 < b4};
:: JGRAPH_6:funcnot 3 => JGRAPH_6:func 2
definition
let a1, a2, a3, a4 be real set;
func closed_inside_of_rectangle(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: a1 <= b1 `1 & b1 `1 <= a2 & a3 <= b1 `2 & b1 `2 <= a4};
end;
:: JGRAPH_6:def 2
theorem
for b1, b2, b3, b4 being real set holds
closed_inside_of_rectangle(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 & b5 `2 <= b4};
:: JGRAPH_6:funcnot 4 => JGRAPH_6:func 3
definition
let a1, a2, a3, a4 be real set;
func outside_of_rectangle(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: (a1 <= b1 `1 & b1 `1 <= a2 & a3 <= b1 `2 implies a4 < b1 `2)};
end;
:: JGRAPH_6:def 3
theorem
for b1, b2, b3, b4 being real set holds
outside_of_rectangle(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: (b1 <= b5 `1 & b5 `1 <= b2 & b3 <= b5 `2 implies b4 < b5 `2)};
:: JGRAPH_6:funcnot 5 => JGRAPH_6:func 4
definition
let a1, a2, a3, a4 be real set;
func closed_outside_of_rectangle(A1,A2,A3,A4) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: (a1 < b1 `1 & b1 `1 < a2 & a3 < b1 `2 implies a4 <= b1 `2)};
end;
:: JGRAPH_6:def 4
theorem
for b1, b2, b3, b4 being real set holds
closed_outside_of_rectangle(b1,b2,b3,b4) = {b5 where b5 is Element of the carrier of TOP-REAL 2: (b1 < b5 `1 & b5 `1 < b2 & b3 < b5 `2 implies b4 <= b5 `2)};
:: JGRAPH_6:th 29
theorem
for b1, b2, b3 being real set
for b4, b5 being Element of bool the carrier of TOP-REAL 2
st 0 <= b3 &
b4 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6.| = 1} &
b5 = {b6 where b6 is Element of the carrier of TOP-REAL 2: |.b6 - |[b1,b2]|.| = b3}
holds (AffineMap(b3,b1,b3,b2)) .: b4 = b5;
:: JGRAPH_6:th 30
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st (ex b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2 st
b3 is being_homeomorphism((TOP-REAL 2) | b1, (TOP-REAL 2) | b2)) &
b1 is being_simple_closed_curve
holds b2 is being_simple_closed_curve;
:: JGRAPH_6:th 31
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st b1 is being_simple_closed_curve
holds b1 is compact(TOP-REAL 2);
:: JGRAPH_6:th 32
theorem
for b1, b2, b3 being real set
for b4 being Element of bool the carrier of TOP-REAL 2
st 0 < b3 &
b4 = {b5 where b5 is Element of the carrier of TOP-REAL 2: |.b5 - |[b1,b2]|.| = b3}
holds b4 is being_simple_closed_curve;
:: JGRAPH_6:funcnot 6 => JGRAPH_6:func 5
definition
let a1, a2, a3 be real set;
func circle(A1,A2,A3) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: |.b1 - |[a1,a2]|.| = a3};
end;
:: JGRAPH_6:def 5
theorem
for b1, b2, b3 being real set holds
circle(b1,b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4 - |[b1,b2]|.| = b3};
:: JGRAPH_6:funcreg 1
registration
let a1, a2, a3 be real set;
cluster circle(a1,a2,a3) -> compact;
end;
:: JGRAPH_6:funcreg 2
registration
let a1, a2 be real set;
let a3 be real non negative set;
cluster circle(a1,a2,a3) -> non empty;
end;
:: JGRAPH_6:funcnot 7 => JGRAPH_6:func 6
definition
let a1, a2, a3 be real set;
func inside_of_circle(A1,A2,A3) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: |.b1 - |[a1,a2]|.| < a3};
end;
:: JGRAPH_6:def 6
theorem
for b1, b2, b3 being real set holds
inside_of_circle(b1,b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4 - |[b1,b2]|.| < b3};
:: JGRAPH_6:funcnot 8 => JGRAPH_6:func 7
definition
let a1, a2, a3 be real set;
func closed_inside_of_circle(A1,A2,A3) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: |.b1 - |[a1,a2]|.| <= a3};
end;
:: JGRAPH_6:def 7
theorem
for b1, b2, b3 being real set holds
closed_inside_of_circle(b1,b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL 2: |.b4 - |[b1,b2]|.| <= b3};
:: JGRAPH_6:funcnot 9 => JGRAPH_6:func 8
definition
let a1, a2, a3 be real set;
func outside_of_circle(A1,A2,A3) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: a3 < |.b1 - |[a1,a2]|.|};
end;
:: JGRAPH_6:def 8
theorem
for b1, b2, b3 being real set holds
outside_of_circle(b1,b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL 2: b3 < |.b4 - |[b1,b2]|.|};
:: JGRAPH_6:funcnot 10 => JGRAPH_6:func 9
definition
let a1, a2, a3 be real set;
func closed_outside_of_circle(A1,A2,A3) -> Element of bool the carrier of TOP-REAL 2 equals
{b1 where b1 is Element of the carrier of TOP-REAL 2: a3 <= |.b1 - |[a1,a2]|.|};
end;
:: JGRAPH_6:def 9
theorem
for b1, b2, b3 being real set holds
closed_outside_of_circle(b1,b2,b3) = {b4 where b4 is Element of the carrier of TOP-REAL 2: b3 <= |.b4 - |[b1,b2]|.|};
:: JGRAPH_6:th 33
theorem
for b1 being real set holds
inside_of_circle(0,0,b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| < b1} &
(b1 <= 0 or circle(0,0,b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| = b1}) &
outside_of_circle(0,0,b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b1 < |.b2.|} &
closed_inside_of_circle(0,0,b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: |.b2.| <= b1} &
closed_outside_of_circle(0,0,b1) = {b2 where b2 is Element of the carrier of TOP-REAL 2: b1 <= |.b2.|};
:: JGRAPH_6:th 34
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: - 1 < b3 `1 & b3 `1 < 1 & - 1 < b3 `2 & b3 `2 < 1} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: |.b3.| < 1}
holds Sq_Circ .: b1 = b2;
:: JGRAPH_6:th 35
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (- 1 <= b3 `1 & b3 `1 <= 1 & - 1 <= b3 `2 implies 1 < b3 `2)} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: 1 < |.b3.|}
holds Sq_Circ .: b1 = b2;
:: JGRAPH_6:th 36
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: - 1 <= b3 `1 & b3 `1 <= 1 & - 1 <= b3 `2 & b3 `2 <= 1} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: |.b3.| <= 1}
holds Sq_Circ .: b1 = b2;
:: JGRAPH_6:th 37
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 = {b3 where b3 is Element of the carrier of TOP-REAL 2: (- 1 < b3 `1 & b3 `1 < 1 & - 1 < b3 `2 implies 1 <= b3 `2)} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: 1 <= |.b3.|}
holds Sq_Circ .: b1 = b2;
:: JGRAPH_6:th 38
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of bool the carrier of TOP-REAL 2
for b9 being non empty compact Element of bool the carrier of TOP-REAL 2
for b10 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b9 = circle(0,0,1) &
b1 = inside_of_circle(0,0,1) &
b2 = outside_of_circle(0,0,1) &
b3 = closed_inside_of_circle(0,0,1) &
b4 = closed_outside_of_circle(0,0,1) &
b5 = inside_of_rectangle(- 1,1,- 1,1) &
b6 = outside_of_rectangle(- 1,1,- 1,1) &
b7 = closed_inside_of_rectangle(- 1,1,- 1,1) &
b8 = closed_outside_of_rectangle(- 1,1,- 1,1) &
b10 = Sq_Circ
holds b10 .: [.- 1,1,- 1,1.] = b9 &
b10 " .: b9 = [.- 1,1,- 1,1.] &
b10 .: b5 = b1 &
b10 " .: b1 = b5 &
b10 .: b6 = b2 &
b10 " .: b2 = b6 &
b10 .: b7 = b3 &
b10 .: b8 = b4 &
b10 " .: b3 = b7 &
b10 " .: b4 = b8;
:: JGRAPH_6:th 39
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds LSeg(|[b1,b3]|,|[b1,b4]|) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 = b1 & b5 `2 <= b4 & b3 <= b5 `2} &
LSeg(|[b1,b4]|,|[b2,b4]|) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= b2 & b1 <= b5 `1 & b5 `2 = b4} &
LSeg(|[b1,b3]|,|[b2,b3]|) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 <= b2 & b1 <= b5 `1 & b5 `2 = b3} &
LSeg(|[b2,b3]|,|[b2,b4]|) = {b5 where b5 is Element of the carrier of TOP-REAL 2: b5 `1 = b2 & b5 `2 <= b4 & b3 <= b5 `2};
:: JGRAPH_6:th 41
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds (LSeg(|[b1,b3]|,|[b1,b4]|)) /\ LSeg(|[b1,b3]|,|[b2,b3]|) = {|[b1,b3]|};
:: JGRAPH_6:th 42
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds (LSeg(|[b1,b3]|,|[b2,b3]|)) /\ LSeg(|[b2,b3]|,|[b2,b4]|) = {|[b2,b3]|};
:: JGRAPH_6:th 43
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds (LSeg(|[b1,b4]|,|[b2,b4]|)) /\ LSeg(|[b2,b3]|,|[b2,b4]|) = {|[b2,b4]|};
:: JGRAPH_6:th 44
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds (LSeg(|[b1,b3]|,|[b1,b4]|)) /\ LSeg(|[b1,b4]|,|[b2,b4]|) = {|[b1,b4]|};
:: JGRAPH_6:th 45
theorem
{b1 where b1 is Element of the carrier of TOP-REAL 2: ((- 1 = b1 `1 & - 1 <= b1 `2 implies 1 < b1 `2) &
(b1 `1 = 1 & - 1 <= b1 `2 implies 1 < b1 `2) &
(- 1 = b1 `2 & - 1 <= b1 `1 implies 1 < b1 `1) implies 1 = b1 `2 & - 1 <= b1 `1 & b1 `1 <= 1)} = {b1 where b1 is Element of the carrier of TOP-REAL 2: ((b1 `1 = - 1 & - 1 <= b1 `2 implies 1 < b1 `2) &
(b1 `2 = 1 & - 1 <= b1 `1 implies 1 < b1 `1) &
(b1 `1 = 1 & - 1 <= b1 `2 implies 1 < b1 `2) implies b1 `2 = - 1 & - 1 <= b1 `1 & b1 `1 <= 1)};
:: JGRAPH_6:th 46
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds W-bound [.b1,b2,b3,b4.] = b1;
:: JGRAPH_6:th 47
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds N-bound [.b1,b2,b3,b4.] = b4;
:: JGRAPH_6:th 48
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds E-bound [.b1,b2,b3,b4.] = b2;
:: JGRAPH_6:th 49
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds S-bound [.b1,b2,b3,b4.] = b3;
:: JGRAPH_6:th 50
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds NW-corner [.b1,b2,b3,b4.] = |[b1,b4]|;
:: JGRAPH_6:th 51
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds NE-corner [.b1,b2,b3,b4.] = |[b2,b4]|;
:: JGRAPH_6:th 52
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds SW-corner [.b1,b2,b3,b4.] = |[b1,b3]|;
:: JGRAPH_6:th 53
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds SE-corner [.b1,b2,b3,b4.] = |[b2,b3]|;
:: JGRAPH_6:th 54
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds W-most [.b1,b2,b3,b4.] = LSeg(|[b1,b3]|,|[b1,b4]|);
:: JGRAPH_6:th 55
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds E-most [.b1,b2,b3,b4.] = LSeg(|[b2,b3]|,|[b2,b4]|);
:: JGRAPH_6:th 56
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds W-min [.b1,b2,b3,b4.] = |[b1,b3]| &
E-max [.b1,b2,b3,b4.] = |[b2,b4]|;
:: JGRAPH_6:th 57
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds (LSeg(|[b1,b3]|,|[b1,b4]|)) \/ LSeg(|[b1,b4]|,|[b2,b4]|) is_an_arc_of W-min [.b1,b2,b3,b4.],E-max [.b1,b2,b3,b4.] &
(LSeg(|[b1,b3]|,|[b2,b3]|)) \/ LSeg(|[b2,b3]|,|[b2,b4]|) is_an_arc_of E-max [.b1,b2,b3,b4.],W-min [.b1,b2,b3,b4.];
:: JGRAPH_6:th 58
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being FinSequence of the carrier of TOP-REAL 2
for b7, b8, b9, b10 being Element of the carrier of TOP-REAL 2
st b1 < b2 & b3 < b4 & b7 = |[b1,b3]| & b8 = |[b2,b4]| & b9 = |[b1,b4]| & b10 = |[b2,b3]| & b5 = <*b7,b9,b8*> & b6 = <*b7,b10,b8*>
holds b5 is being_S-Seq &
L~ b5 = (LSeg(b7,b9)) \/ LSeg(b9,b8) &
b6 is being_S-Seq &
L~ b6 = (LSeg(b7,b10)) \/ LSeg(b10,b8) &
[.b1,b2,b3,b4.] = (L~ b5) \/ L~ b6 &
(L~ b5) /\ L~ b6 = {b7,b8} &
b5 /. 1 = b7 &
b5 /. len b5 = b8 &
b6 /. 1 = b7 &
b6 /. len b6 = b8;
:: JGRAPH_6:th 59
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being real set
for b7, b8 being FinSequence of the carrier of TOP-REAL 2
for b9, b10 being Element of the carrier of TOP-REAL 2
st b3 < b4 &
b5 < b6 &
b9 = |[b3,b5]| &
b10 = |[b4,b6]| &
b7 = <*|[b3,b5]|,|[b3,b6]|,|[b4,b6]|*> &
b8 = <*|[b3,b5]|,|[b4,b5]|,|[b4,b6]|*> &
b1 = L~ b7 &
b2 = L~ b8
holds b1 is_an_arc_of b9,b10 & b2 is_an_arc_of b9,b10 & b1 is not empty & b2 is not empty & [.b3,b4,b5,b6.] = b1 \/ b2 & b1 /\ b2 = {b9,b10};
:: JGRAPH_6:th 60
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds [.b1,b2,b3,b4.] is being_simple_closed_curve;
:: JGRAPH_6:th 61
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds Upper_Arc [.b1,b2,b3,b4.] = (LSeg(|[b1,b3]|,|[b1,b4]|)) \/ LSeg(|[b1,b4]|,|[b2,b4]|);
:: JGRAPH_6:th 62
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds Lower_Arc [.b1,b2,b3,b4.] = (LSeg(|[b1,b3]|,|[b2,b3]|)) \/ LSeg(|[b2,b3]|,|[b2,b4]|);
:: JGRAPH_6:th 63
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds ex b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | Upper_Arc [.b1,b2,b3,b4.] st
b5 is being_homeomorphism(I[01], (TOP-REAL 2) | Upper_Arc [.b1,b2,b3,b4.]) &
b5 . 0 = W-min [.b1,b2,b3,b4.] &
b5 . 1 = E-max [.b1,b2,b3,b4.] &
rng b5 = Upper_Arc [.b1,b2,b3,b4.] &
(for b6 being Element of REAL
st b6 in [.0,1 / 2.]
holds b5 . b6 = ((1 - (2 * b6)) * |[b1,b3]|) + ((2 * b6) * |[b1,b4]|)) &
(for b6 being Element of REAL
st b6 in [.1 / 2,1.]
holds b5 . b6 = ((1 - ((2 * b6) - 1)) * |[b1,b4]|) + (((2 * b6) - 1) * |[b2,b4]|)) &
(for b6 being Element of the carrier of TOP-REAL 2
st b6 in LSeg(|[b1,b3]|,|[b1,b4]|)
holds 0 <= ((b6 `2 - b3) / (b4 - b3)) / 2 &
((b6 `2 - b3) / (b4 - b3)) / 2 <= 1 &
b5 . (((b6 `2 - b3) / (b4 - b3)) / 2) = b6) &
(for b6 being Element of the carrier of TOP-REAL 2
st b6 in LSeg(|[b1,b4]|,|[b2,b4]|)
holds 0 <= (((b6 `1 - b1) / (b2 - b1)) / 2) + (1 / 2) &
(((b6 `1 - b1) / (b2 - b1)) / 2) + (1 / 2) <= 1 &
b5 . ((((b6 `1 - b1) / (b2 - b1)) / 2) + (1 / 2)) = b6);
:: JGRAPH_6:th 64
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds ex b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | Lower_Arc [.b1,b2,b3,b4.] st
b5 is being_homeomorphism(I[01], (TOP-REAL 2) | Lower_Arc [.b1,b2,b3,b4.]) &
b5 . 0 = E-max [.b1,b2,b3,b4.] &
b5 . 1 = W-min [.b1,b2,b3,b4.] &
rng b5 = Lower_Arc [.b1,b2,b3,b4.] &
(for b6 being Element of REAL
st b6 in [.0,1 / 2.]
holds b5 . b6 = ((1 - (2 * b6)) * |[b2,b4]|) + ((2 * b6) * |[b2,b3]|)) &
(for b6 being Element of REAL
st b6 in [.1 / 2,1.]
holds b5 . b6 = ((1 - ((2 * b6) - 1)) * |[b2,b3]|) + (((2 * b6) - 1) * |[b1,b3]|)) &
(for b6 being Element of the carrier of TOP-REAL 2
st b6 in LSeg(|[b2,b4]|,|[b2,b3]|)
holds 0 <= ((b6 `2 - b4) / (b3 - b4)) / 2 &
((b6 `2 - b4) / (b3 - b4)) / 2 <= 1 &
b5 . (((b6 `2 - b4) / (b3 - b4)) / 2) = b6) &
(for b6 being Element of the carrier of TOP-REAL 2
st b6 in LSeg(|[b2,b3]|,|[b1,b3]|)
holds 0 <= (((b6 `1 - b2) / (b1 - b2)) / 2) + (1 / 2) &
(((b6 `1 - b2) / (b1 - b2)) / 2) + (1 / 2) <= 1 &
b5 . ((((b6 `1 - b2) / (b1 - b2)) / 2) + (1 / 2)) = b6);
:: JGRAPH_6:th 65
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b1,b3]|,|[b1,b4]|) &
b6 in LSeg(|[b1,b3]|,|[b1,b4]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
b5 `2 <= b6 `2;
:: JGRAPH_6:th 66
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b1,b4]|,|[b2,b4]|) &
b6 in LSeg(|[b1,b4]|,|[b2,b4]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
b5 `1 <= b6 `1;
:: JGRAPH_6:th 67
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b2,b3]|,|[b2,b4]|) &
b6 in LSeg(|[b2,b3]|,|[b2,b4]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
b6 `2 <= b5 `2;
:: JGRAPH_6:th 68
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b1,b3]|,|[b2,b3]|) &
b6 in LSeg(|[b1,b3]|,|[b2,b3]|)
holds LE b5,b6,[.b1,b2,b3,b4.] & b5 <> W-min [.b1,b2,b3,b4.]
iff
b6 `1 <= b5 `1 & b6 <> W-min [.b1,b2,b3,b4.];
:: JGRAPH_6:th 69
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b1,b3]|,|[b1,b4]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
((b6 in LSeg(|[b1,b3]|,|[b1,b4]|) implies b6 `2 < b5 `2) &
not b6 in LSeg(|[b1,b4]|,|[b2,b4]|) &
not b6 in LSeg(|[b2,b4]|,|[b2,b3]|) implies b6 in LSeg(|[b2,b3]|,|[b1,b3]|) &
b6 <> W-min [.b1,b2,b3,b4.]);
:: JGRAPH_6:th 70
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b1,b4]|,|[b2,b4]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
((b6 in LSeg(|[b1,b4]|,|[b2,b4]|) implies b6 `1 < b5 `1) &
not b6 in LSeg(|[b2,b4]|,|[b2,b3]|) implies b6 in LSeg(|[b2,b3]|,|[b1,b3]|) &
b6 <> W-min [.b1,b2,b3,b4.]);
:: JGRAPH_6:th 71
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b2,b4]|,|[b2,b3]|)
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
(b6 in LSeg(|[b2,b4]|,|[b2,b3]|) &
b6 `2 <= b5 `2 or b6 in LSeg(|[b2,b3]|,|[b1,b3]|) &
b6 <> W-min [.b1,b2,b3,b4.]);
:: JGRAPH_6:th 72
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
st b1 < b2 &
b3 < b4 &
b5 in LSeg(|[b2,b3]|,|[b1,b3]|) &
b5 <> W-min [.b1,b2,b3,b4.]
holds LE b5,b6,[.b1,b2,b3,b4.]
iff
b6 in LSeg(|[b2,b3]|,|[b1,b3]|) &
b6 `1 <= b5 `1 &
b6 <> W-min [.b1,b2,b3,b4.];
:: JGRAPH_6:th 73
theorem
for b1 being set
for b2, b3, b4, b5 being real set
st b1 in [.b2,b3,b4,b5.] &
b2 < b3 &
b4 < b5 &
not b1 in LSeg(|[b2,b4]|,|[b2,b5]|) &
not b1 in LSeg(|[b2,b5]|,|[b3,b5]|) &
not b1 in LSeg(|[b3,b5]|,|[b3,b4]|)
holds b1 in LSeg(|[b3,b4]|,|[b2,b4]|);
:: JGRAPH_6:th 74
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st LE b1,b2,[.- 1,1,- 1,1.] &
b1 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
(b2 in LSeg(|[- 1,- 1]|,|[- 1,1]|) implies b2 `2 < b1 `2) &
not b2 in LSeg(|[- 1,1]|,|[1,1]|) &
not b2 in LSeg(|[1,1]|,|[1,- 1]|)
holds b2 in LSeg(|[1,- 1]|,|[- 1,- 1]|) &
b2 <> |[- 1,- 1]|;
:: JGRAPH_6:th 75
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
for b4 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b3 = circle(0,0,1) &
b4 = Sq_Circ &
b1 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
0 <= b1 `2 &
LE b1,b2,[.- 1,1,- 1,1.]
holds LE b4 . b1,b4 . b2,b3;
:: JGRAPH_6:th 76
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being non empty compact Element of bool the carrier of TOP-REAL 2
for b5 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b4 = circle(0,0,1) &
b5 = Sq_Circ &
b1 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
0 <= b1 `2 &
LE b1,b2,[.- 1,1,- 1,1.] &
LE b2,b3,[.- 1,1,- 1,1.]
holds LE b5 . b2,b5 . b3,b4;
:: JGRAPH_6:th 77
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b2 = Sq_Circ & b1 `1 = - 1 & b1 `2 < 0
holds (b2 . b1) `1 < 0 & (b2 . b1) `2 < 0;
:: JGRAPH_6:th 78
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty compact Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b2 = circle(0,0,1) & b3 = Sq_Circ
holds 0 <= (b3 . b1) `1
iff
0 <= b1 `1;
:: JGRAPH_6:th 79
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being non empty compact Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b2 = circle(0,0,1) & b3 = Sq_Circ
holds 0 <= (b3 . b1) `2
iff
0 <= b1 `2;
:: JGRAPH_6:th 80
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b3 = Sq_Circ &
b1 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
b2 in LSeg(|[1,- 1]|,|[- 1,- 1]|)
holds (b3 . b1) `1 <= (b3 . b2) `1;
:: JGRAPH_6:th 81
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b3 = Sq_Circ &
b1 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
b2 in LSeg(|[- 1,- 1]|,|[- 1,1]|) &
b2 `2 <= b1 `2 &
b1 `2 < 0
holds (b3 . b2) `2 <= (b3 . b1) `2;
:: JGRAPH_6:th 82
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b5 = circle(0,0,1) &
b6 = Sq_Circ &
LE b1,b2,[.- 1,1,- 1,1.] &
LE b2,b3,[.- 1,1,- 1,1.] &
LE b3,b4,[.- 1,1,- 1,1.]
holds b6 . b1,b6 . b2,b6 . b3,b6 . b4 are_in_this_order_on b5;
:: JGRAPH_6:th 83
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being non empty compact Element of bool the carrier of TOP-REAL 2
st b3 is being_simple_closed_curve & b1 in b3 & b2 in b3 & not LE b1,b2,b3
holds LE b2,b1,b3;
:: JGRAPH_6:th 84
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being non empty compact Element of bool the carrier of TOP-REAL 2
st b4 is being_simple_closed_curve & b1 in b4 & b2 in b4 & b3 in b4 & (LE b1,b2,b4 implies not LE b2,b3,b4) & (LE b1,b3,b4 implies not LE b3,b2,b4) & (LE b2,b1,b4 implies not LE b1,b3,b4) & (LE b2,b3,b4 implies not LE b3,b1,b4) & (LE b3,b1,b4 implies not LE b1,b2,b4)
holds LE b3,b2,b4 & LE b2,b1,b4;
:: JGRAPH_6:th 85
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being non empty compact Element of bool the carrier of TOP-REAL 2
st b4 is being_simple_closed_curve & b1 in b4 & b2 in b4 & b3 in b4 & LE b2,b3,b4 & not LE b1,b2,b4 & (LE b2,b1,b4 implies not LE b1,b3,b4)
holds LE b3,b1,b4;
:: JGRAPH_6:th 86
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
st b5 is being_simple_closed_curve & b1 in b5 & b2 in b5 & b3 in b5 & b4 in b5 & LE b2,b3,b5 & LE b3,b4,b5 & not LE b1,b2,b5 & (LE b2,b1,b5 implies not LE b1,b3,b5) & (LE b3,b1,b5 implies not LE b1,b4,b5)
holds LE b4,b1,b5;
:: JGRAPH_6:th 87
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b5 = circle(0,0,1) & b6 = Sq_Circ & LE b6 . b1,b6 . b2,b5 & LE b6 . b2,b6 . b3,b5 & LE b6 . b3,b6 . b4,b5
holds b1,b2,b3,b4 are_in_this_order_on [.- 1,1,- 1,1.];
:: JGRAPH_6:th 88
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being non empty compact Element of bool the carrier of TOP-REAL 2
for b6 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b5 = circle(0,0,1) & b6 = Sq_Circ
holds b1,b2,b3,b4 are_in_this_order_on [.- 1,1,- 1,1.]
iff
b6 . b1,b6 . b2,b6 . b3,b6 . b4 are_in_this_order_on b5;
:: JGRAPH_6:th 89
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
for b5 being Element of bool the carrier of TOP-REAL 2
st b1,b2,b3,b4 are_in_this_order_on [.- 1,1,- 1,1.]
for b6, b7 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL 2
st b6 is continuous(I[01], TOP-REAL 2) &
b6 is one-to-one &
b7 is continuous(I[01], TOP-REAL 2) &
b7 is one-to-one &
b5 = closed_inside_of_rectangle(- 1,1,- 1,1) &
b6 . 0 = b1 &
b6 . 1 = b3 &
b7 . 0 = b2 &
b7 . 1 = b4 &
rng b6 c= b5 &
rng b7 c= b5
holds rng b6 meets rng b7;