Article JGRAPH_3, MML version 4.99.1005
:: JGRAPH_3:th 10
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
|.b1.| = sqrt (b1 `1 ^2 + (b1 `2 ^2)) &
|.b1.| ^2 = b1 `1 ^2 + (b1 `2 ^2);
:: JGRAPH_3:th 11
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set holds
(b1 | b2) .: b3 = b1 .: (b3 /\ b2);
:: JGRAPH_3:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being non empty Element of bool the carrier of b1
for b5 being non empty Element of bool the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b4 ` = {b3} &
b5 ` = {b6 . b3} &
b1 is being_T2 &
b2 is being_T2 &
(for b7 being Element of the carrier of b1 | b4 holds
b6 . b7 <> b6 . b3) &
b6 | b4 is Function-like quasi_total continuous Relation of the carrier of b1 | b4,the carrier of b2 | b5 &
(for b7 being Element of bool the carrier of b2
st b6 . b3 in b7 & b7 is open(b2)
holds ex b8 being Element of bool the carrier of b1 st
b3 in b8 & b8 is open(b1) & b6 .: b8 c= b7)
holds b6 is continuous(b1, b2);
:: JGRAPH_3:funcnot 1 => JGRAPH_3:func 1
definition
func Sq_Circ -> Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 means
for b1 being Element of the carrier of TOP-REAL 2 holds
(b1 = 0.REAL 2 implies it . b1 = b1) &
((b1 `2 <= b1 `1 & - (b1 `1) <= b1 `2 or b1 `1 <= b1 `2 & b1 `2 <= - (b1 `1)) &
b1 <> 0.REAL 2 implies it . b1 = |[b1 `1 / sqrt (1 + ((b1 `2 / (b1 `1)) ^2)),b1 `2 / sqrt (1 + ((b1 `2 / (b1 `1)) ^2))]|) &
((b1 `2 <= b1 `1 implies b1 `2 < - (b1 `1)) &
(b1 `1 <= b1 `2 implies - (b1 `1) < b1 `2) &
b1 <> 0.REAL 2 implies it . b1 = |[b1 `1 / sqrt (1 + ((b1 `1 / (b1 `2)) ^2)),b1 `2 / sqrt (1 + ((b1 `1 / (b1 `2)) ^2))]|);
end;
:: JGRAPH_3:def 1
theorem
for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 holds
b1 = Sq_Circ
iff
for b2 being Element of the carrier of TOP-REAL 2 holds
(b2 = 0.REAL 2 implies b1 . b2 = b2) &
((b2 `2 <= b2 `1 & - (b2 `1) <= b2 `2 or b2 `1 <= b2 `2 & b2 `2 <= - (b2 `1)) &
b2 <> 0.REAL 2 implies b1 . b2 = |[b2 `1 / sqrt (1 + ((b2 `2 / (b2 `1)) ^2)),b2 `2 / sqrt (1 + ((b2 `2 / (b2 `1)) ^2))]|) &
((b2 `2 <= b2 `1 implies b2 `2 < - (b2 `1)) &
(b2 `1 <= b2 `2 implies - (b2 `1) < b2 `2) &
b2 <> 0.REAL 2 implies b1 . b2 = |[b2 `1 / sqrt (1 + ((b2 `1 / (b2 `2)) ^2)),b2 `2 / sqrt (1 + ((b2 `1 / (b2 `2)) ^2))]|);
:: JGRAPH_3:th 14
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2
holds ((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
(b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) or Sq_Circ . b1 = |[b1 `1 / sqrt (1 + ((b1 `1 / (b1 `2)) ^2)),b1 `2 / sqrt (1 + ((b1 `1 / (b1 `2)) ^2))]|) &
((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
(b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) implies Sq_Circ . b1 = |[b1 `1 / sqrt (1 + ((b1 `2 / (b1 `1)) ^2)),b1 `2 / sqrt (1 + ((b1 `2 / (b1 `1)) ^2))]|);
:: JGRAPH_3: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
st b2 is continuous(b1, R^1) &
(for b3 being Element of the carrier of b1 holds
ex b4 being real set st
b2 . b3 = b4 & 0 <= b4)
holds ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1 st
(for b4 being Element of the carrier of b1
for b5 being real set
st b2 . b4 = b5
holds b3 . b4 = sqrt b5) &
b3 is continuous(b1, R^1);
:: JGRAPH_3:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = (b6 / b7) ^2) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = 1 + ((b6 / b7) ^2)) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = sqrt (1 + ((b6 / b7) ^2))) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = b6 / sqrt (1 + ((b6 / b7) ^2))) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = b7 / sqrt (1 + ((b6 / b7) ^2))) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 21
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `1 / sqrt (1 + ((b3 `2 / (b3 `1)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `1 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 22
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `2 / sqrt (1 + ((b3 `2 / (b3 `1)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `1 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 23
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `2 / sqrt (1 + ((b3 `1 / (b3 `2)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `2 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 24
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `1 / sqrt (1 + ((b3 `1 / (b3 `2)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `2 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 25
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
st b3 = Sq_Circ | b1 &
b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
b4 <> 0.REAL 2}
holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);
:: JGRAPH_3:th 26
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
st b3 = Sq_Circ | b1 &
b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
b4 <> 0.REAL 2}
holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);
:: JGRAPH_3:sch 1
scheme JGRAPH_3:sch 1
{b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1] & b1 <> 0.REAL 2} c= (the carrier of TOP-REAL 2) \ {0.REAL 2}
:: JGRAPH_3:sch 2
scheme JGRAPH_3:sch 2
{b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1] & b1 <> 0.REAL 2} = {b1 where b1 is Element of the carrier of TOP-REAL 2: P1[b1]} /\ ((the carrier of TOP-REAL 2) \ {0.REAL 2})
:: JGRAPH_3:th 27
theorem
for b1 being 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 ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
st b3 = Sq_Circ | b2 &
b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
b4 <> 0.REAL 2}
holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
b2 is closed((TOP-REAL 2) | b1);
:: JGRAPH_3:th 28
theorem
for b1 being 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 ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
st b3 = Sq_Circ | b2 &
b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
b4 <> 0.REAL 2}
holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
b2 is closed((TOP-REAL 2) | b1);
:: JGRAPH_3:th 29
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
st b1 ` = {0.REAL 2}
holds ex b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b1 st
b2 = Sq_Circ | b1 &
b2 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b1);
:: JGRAPH_3:th 30
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
st b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2}
holds b1 ` = {0.REAL 2};
:: JGRAPH_3:th 31
theorem
ex b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
b1 = Sq_Circ & b1 is continuous(TOP-REAL 2, TOP-REAL 2);
:: JGRAPH_3:th 32
theorem
Sq_Circ is one-to-one;
:: JGRAPH_3:funcreg 1
registration
cluster Sq_Circ -> Function-like one-to-one quasi_total;
end;
:: JGRAPH_3:th 33
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 & - 1 <= b3 `2 implies 1 < b3 `2) &
(b3 `1 = 1 & - 1 <= b3 `2 implies 1 < b3 `2) &
(- 1 = b3 `2 & - 1 <= b3 `1 implies 1 < b3 `1) implies 1 = b3 `2 & - 1 <= b3 `1 & b3 `1 <= 1)} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: |.b3.| = 1}
holds Sq_Circ .: b1 = b2;
:: JGRAPH_3:th 34
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b2,the carrier of (TOP-REAL 2) | b1
st b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: ((- 1 = b4 `1 & - 1 <= b4 `2 implies 1 < b4 `2) &
(b4 `1 = 1 & - 1 <= b4 `2 implies 1 < b4 `2) &
(- 1 = b4 `2 & - 1 <= b4 `1 implies 1 < b4 `1) implies 1 = b4 `2 & - 1 <= b4 `1 & b4 `1 <= 1)} &
b3 is being_homeomorphism((TOP-REAL 2) | b2, (TOP-REAL 2) | b1)
holds b1 is being_simple_closed_curve;
:: JGRAPH_3:th 35
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: ((- 1 = b2 `1 & - 1 <= b2 `2 implies 1 < b2 `2) &
(b2 `1 = 1 & - 1 <= b2 `2 implies 1 < b2 `2) &
(- 1 = b2 `2 & - 1 <= b2 `1 implies 1 < b2 `1) implies 1 = b2 `2 & - 1 <= b2 `1 & b2 `1 <= 1)}
holds b1 is being_simple_closed_curve & b1 is compact(TOP-REAL 2);
:: JGRAPH_3:th 36
theorem
for b1 being 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 b1 is being_simple_closed_curve;
:: JGRAPH_3: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 & b3 `2 <= 1} &
b2 = {b3 where b3 is Element of the carrier of TOP-REAL 2: |.b3.| <= 1}
holds Sq_Circ " b2 c= b1;
:: JGRAPH_3:th 38
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
(b1 = 0.REAL 2 implies Sq_Circ " . b1 = 0.REAL 2) &
((b1 `2 <= b1 `1 & - (b1 `1) <= b1 `2 or b1 `1 <= b1 `2 & b1 `2 <= - (b1 `1)) &
b1 <> 0.REAL 2 implies Sq_Circ " . b1 = |[b1 `1 * sqrt (1 + ((b1 `2 / (b1 `1)) ^2)),b1 `2 * sqrt (1 + ((b1 `2 / (b1 `1)) ^2))]|) &
((b1 `2 <= b1 `1 implies b1 `2 < - (b1 `1)) &
(b1 `1 <= b1 `2 implies - (b1 `1) < b1 `2) &
b1 <> 0.REAL 2 implies Sq_Circ " . b1 = |[b1 `1 * sqrt (1 + ((b1 `1 / (b1 `2)) ^2)),b1 `2 * sqrt (1 + ((b1 `1 / (b1 `2)) ^2))]|);
:: JGRAPH_3:th 39
theorem
Sq_Circ " is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2;
:: JGRAPH_3:th 40
theorem
for b1 being Element of the carrier of TOP-REAL 2
st b1 <> 0.REAL 2
holds ((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
(b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) or Sq_Circ " . b1 = |[b1 `1 * sqrt (1 + ((b1 `1 / (b1 `2)) ^2)),b1 `2 * sqrt (1 + ((b1 `1 / (b1 `2)) ^2))]|) &
((b1 `1 <= b1 `2 implies b1 `1 < - (b1 `2)) &
(b1 `2 <= b1 `1 implies - (b1 `2) < b1 `1) implies Sq_Circ " . b1 = |[b1 `1 * sqrt (1 + ((b1 `2 / (b1 `1)) ^2)),b1 `2 * sqrt (1 + ((b1 `2 / (b1 `1)) ^2))]|);
:: JGRAPH_3:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = b6 * sqrt (1 + ((b6 / b7) ^2))) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
st b2 is continuous(b1, R^1) &
b3 is continuous(b1, R^1) &
(for b4 being Element of the carrier of b1 holds
b3 . b4 <> 0)
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, b7 being real set
st b2 . b5 = b6 & b3 . b5 = b7
holds b4 . b5 = b7 * sqrt (1 + ((b6 / b7) ^2))) &
b4 is continuous(b1, R^1);
:: JGRAPH_3:th 43
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `1 * sqrt (1 + ((b3 `2 / (b3 `1)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `1 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 44
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `2 * sqrt (1 + ((b3 `2 / (b3 `1)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `1 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 45
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `2 * sqrt (1 + ((b3 `1 / (b3 `2)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `2 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 46
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of R^1
st (for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b2 . b3 = b3 `1 * sqrt (1 + ((b3 `1 / (b3 `2)) ^2))) &
(for b3 being Element of the carrier of TOP-REAL 2
st b3 in the carrier of (TOP-REAL 2) | b1
holds b3 `2 <> 0)
holds b2 is continuous((TOP-REAL 2) | b1, R^1);
:: JGRAPH_3:th 47
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
st b3 = Sq_Circ " | b1 &
b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
b4 <> 0.REAL 2}
holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);
:: JGRAPH_3:th 48
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b2
st b3 = Sq_Circ " | b1 &
b2 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b1 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
b4 <> 0.REAL 2}
holds b3 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b2);
:: JGRAPH_3:th 49
theorem
for b1 being 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 ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
st b3 = Sq_Circ " | b2 &
b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `2 <= b4 `1 & - (b4 `1) <= b4 `2 or b4 `1 <= b4 `2 & b4 `2 <= - (b4 `1)) &
b4 <> 0.REAL 2}
holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
b2 is closed((TOP-REAL 2) | b1);
:: JGRAPH_3:th 50
theorem
for b1 being 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 ((TOP-REAL 2) | b1) | b2,the carrier of (TOP-REAL 2) | b1
st b3 = Sq_Circ " | b2 &
b1 = (the carrier of TOP-REAL 2) \ {0.REAL 2} &
b2 = {b4 where b4 is Element of the carrier of TOP-REAL 2: (b4 `1 <= b4 `2 & - (b4 `2) <= b4 `1 or b4 `2 <= b4 `1 & b4 `1 <= - (b4 `2)) &
b4 <> 0.REAL 2}
holds b3 is continuous(((TOP-REAL 2) | b1) | b2, (TOP-REAL 2) | b1) &
b2 is closed((TOP-REAL 2) | b1);
:: JGRAPH_3:th 51
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
st b1 ` = {0.REAL 2}
holds ex b2 being Function-like quasi_total Relation of the carrier of (TOP-REAL 2) | b1,the carrier of (TOP-REAL 2) | b1 st
b2 = Sq_Circ " | b1 &
b2 is continuous((TOP-REAL 2) | b1, (TOP-REAL 2) | b1);
:: JGRAPH_3:th 52
theorem
ex b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 st
b1 = Sq_Circ " & b1 is continuous(TOP-REAL 2, TOP-REAL 2);
:: JGRAPH_3:th 54
theorem
Sq_Circ is Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2 &
rng Sq_Circ = the carrier of TOP-REAL 2 &
(for b1 being Function-like quasi_total Relation of the carrier of TOP-REAL 2,the carrier of TOP-REAL 2
st b1 = Sq_Circ
holds b1 is being_homeomorphism(TOP-REAL 2, TOP-REAL 2));
:: JGRAPH_3:th 55
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 b5 &
b1 . b9 in b4 &
b2 . b8 in b7 &
b2 . b9 in b6 &
rng b1 c= b3 &
rng b2 c= b3
holds rng b1 meets rng b2;