Article JORDAN, MML version 4.99.1005
:: JORDAN:funcreg 1
registration
let a1 be Reflexive symmetric triangle MetrStruct;
let a2, a3 be Element of the carrier of a1;
cluster dist(a2,a3) -> non negative;
end;
:: JORDAN:funcreg 2
registration
let a1 be Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
cluster dist(a2,a3) -> non negative;
end;
:: JORDAN:th 1
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2 <> b3
holds (1 / 2) * (b2 + b3) <> b2;
:: JORDAN:th 2
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st b1 `2 < b2 `2
holds b1 `2 < ((1 / 2) * (b1 + b2)) `2;
:: JORDAN:th 3
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
st b1 `2 < b2 `2
holds ((1 / 2) * (b1 + b2)) `2 < b2 `2;
:: JORDAN:th 4
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being vertical Element of bool the carrier of TOP-REAL 2 holds
b2 /\ b1 is vertical;
:: JORDAN:th 5
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2 being horizontal Element of bool the carrier of TOP-REAL 2 holds
b2 /\ b1 is horizontal;
:: JORDAN:th 6
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 in LSeg(b2,b3) & LSeg(b2,b3) is vertical
holds LSeg(b1,b3) is vertical;
:: JORDAN:th 7
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b1 in LSeg(b2,b3) & LSeg(b2,b3) is horizontal
holds LSeg(b1,b3) is horizontal;
:: JORDAN:funcreg 3
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(SW-corner a1,SE-corner a1) -> horizontal;
end;
:: JORDAN:funcreg 4
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(NW-corner a1,SW-corner a1) -> vertical;
end;
:: JORDAN:funcreg 5
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(NE-corner a1,SE-corner a1) -> vertical;
end;
:: JORDAN:funcreg 6
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(SE-corner a1,SW-corner a1) -> horizontal;
end;
:: JORDAN:funcreg 7
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(SW-corner a1,NW-corner a1) -> vertical;
end;
:: JORDAN:funcreg 8
registration
let a1 be Element of bool the carrier of TOP-REAL 2;
cluster LSeg(SE-corner a1,NE-corner a1) -> vertical;
end;
:: JORDAN:condreg 1
registration
cluster non empty compact vertical -> with_the_max_arc (Element of bool the carrier of TOP-REAL 2);
end;
:: JORDAN:th 8
theorem
for b1 being real set
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 `1 <= b1 & b1 <= b3 `1
holds LSeg(b2,b3) meets Vertical_Line b1;
:: JORDAN:th 9
theorem
for b1 being real set
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 `2 <= b1 & b1 <= b3 `2
holds LSeg(b2,b3) meets Horizontal_Line b1;
:: JORDAN:condreg 2
registration
let a1 be Element of NAT;
cluster empty -> Bounded (Element of bool the carrier of TOP-REAL a1);
end;
:: JORDAN:condreg 3
registration
let a1 be Element of NAT;
cluster non Bounded -> non empty (Element of bool the carrier of TOP-REAL a1);
end;
:: JORDAN:exreg 1
registration
let a1 be non empty Element of NAT;
cluster open closed convex non Bounded Element of bool the carrier of TOP-REAL a1;
end;
:: JORDAN:th 10
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
(north_halfline UMP b1) \ {UMP b1} misses b1;
:: JORDAN:th 11
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
(south_halfline LMP b1) \ {LMP b1} misses b1;
:: JORDAN:th 12
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
(north_halfline UMP b1) \ {UMP b1} c= UBD b1;
:: JORDAN:th 13
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
(south_halfline LMP b1) \ {LMP b1} c= UBD b1;
:: JORDAN:th 14
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 is_inside_component_of b2
holds UBD b2 misses b1;
:: JORDAN:th 15
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 is_outside_component_of b2
holds BDD b2 misses b1;
:: JORDAN:th 16
theorem
for b1 being Element of NAT
for b2 being positive real set
for b3 being Element of the carrier of TOP-REAL b1 holds
b3 in Ball(b3,b2);
:: JORDAN:th 17
theorem
for b1 being Element of NAT
for b2 being non negative real set
for b3 being Element of the carrier of TOP-REAL b1 holds
b3 is Element of the carrier of Tdisk(b3,b2);
:: JORDAN:funcreg 9
registration
let a1 be positive real set;
let a2 be non empty Element of NAT;
let a3, a4 be Element of the carrier of TOP-REAL a2;
cluster (cl_Ball(a3,a1)) \ {a4} -> non empty;
end;
:: JORDAN:th 18
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4 being Element of the carrier of TOP-REAL b3
st b1 <= b2
holds Ball(b4,b1) c= Ball(b4,b2);
:: JORDAN:th 19
theorem
for b1 being real set
for b2 being Element of NAT
for b3 being Element of the carrier of TOP-REAL b2 holds
(cl_Ball(b3,b1)) \ Ball(b3,b1) = Sphere(b3,b1);
:: JORDAN:th 20
theorem
for b1 being real set
for b2 being Element of NAT
for b3, b4 being Element of the carrier of TOP-REAL b2
st b3 in Sphere(b4,b1)
holds (LSeg(b4,b3)) \ {b4,b3} c= Ball(b4,b1);
:: JORDAN:th 21
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4 being Element of the carrier of TOP-REAL b3
st b1 < b2
holds cl_Ball(b4,b1) c= Ball(b4,b2);
:: JORDAN:th 22
theorem
for b1, b2 being real set
for b3 being Element of NAT
for b4 being Element of the carrier of TOP-REAL b3
st b1 < b2
holds Sphere(b4,b1) c= Ball(b4,b2);
:: JORDAN:th 23
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being non empty real set holds
Cl Ball(b2,b3) = cl_Ball(b2,b3);
:: JORDAN:th 24
theorem
for b1 being Element of NAT
for b2 being Element of the carrier of TOP-REAL b1
for b3 being non empty real set holds
Fr Ball(b2,b3) = Sphere(b2,b3);
:: JORDAN:condreg 4
registration
let a1 be non empty Element of NAT;
cluster Bounded -> proper (Element of bool the carrier of TOP-REAL a1);
end;
:: JORDAN:exreg 2
registration
let a1 be Element of NAT;
cluster non empty closed convex Bounded Element of bool the carrier of TOP-REAL a1;
end;
:: JORDAN:exreg 3
registration
let a1 be Element of NAT;
cluster non empty open convex Bounded Element of bool the carrier of TOP-REAL a1;
end;
:: JORDAN:funcreg 10
registration
let a1 be Element of NAT;
let a2 be Bounded Element of bool the carrier of TOP-REAL a1;
cluster Cl a2 -> Bounded;
end;
:: JORDAN:funcreg 11
registration
let a1 be Element of NAT;
let a2 be Bounded Element of bool the carrier of TOP-REAL a1;
cluster Fr a2 -> Bounded;
end;
:: JORDAN:th 25
theorem
for b1 being Element of NAT
for b2 being closed Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1
st not b3 in b2
holds ex b4 being positive real set st
Ball(b3,b4) misses b2;
:: JORDAN:th 26
theorem
for b1 being Element of NAT
for b2 being Bounded Element of bool the carrier of TOP-REAL b1
for b3 being Element of the carrier of TOP-REAL b1 holds
ex b4 being positive real set st
b2 c= Ball(b3,b4);
:: JORDAN:th 27
theorem
for b1, b2 being TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is being_homeomorphism(b1, b2)
holds b3 is onto(the carrier of b1, the carrier of b2);
:: JORDAN:condreg 5
registration
let a1 be non empty TopSpace-like being_T2 TopStruct;
cluster non empty -> being_T2 (SubSpace of a1);
end;
:: JORDAN:funcreg 12
registration
let a1 be Element of the carrier of TOP-REAL 2;
let a2 be real set;
cluster Tdisk(a1,a2) -> closed;
end;
:: JORDAN:funcreg 13
registration
let a1 be Element of the carrier of TOP-REAL 2;
let a2 be real set;
cluster Tdisk(a1,a2) -> compact;
end;
:: JORDAN:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds rng b4 is connected(b1);
:: JORDAN:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7 being Path of b3,b4
st b3 = b5 & b4 = b6 & b3,b4 are_connected & rng b7 c= the carrier of b2
holds b5,b6 are_connected & b7 is Path of b5,b6;
:: JORDAN:th 30
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being Element of the carrier of b1
for b5, b6 being Element of the carrier of b2
for b7 being Path of b3,b4
st b3 = b5 & b4 = b6 & rng b7 c= the carrier of b2
holds b5,b6 are_connected & b7 is Path of b5,b6;
:: JORDAN:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3
st b2,b3 are_connected
holds rng b4 = rng - b4;
:: JORDAN:th 32
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3 being Element of the carrier of b1
for b4 being Path of b2,b3 holds
rng b4 = rng - b4;
:: JORDAN:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4
st b2,b3 are_connected & b3,b4 are_connected
holds rng b5 c= rng (b5 + b6);
:: JORDAN:th 34
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4 holds
rng b5 c= rng (b5 + b6);
:: JORDAN:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b3,b4
for b6 being Path of b2,b3
st b2,b3 are_connected & b3,b4 are_connected
holds rng b5 c= rng (b6 + b5);
:: JORDAN:th 36
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b3,b4
for b6 being Path of b2,b3 holds
rng b5 c= rng (b6 + b5);
:: JORDAN:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4
st b2,b3 are_connected & b3,b4 are_connected
holds rng (b5 + b6) = (rng b5) \/ rng b6;
:: JORDAN:th 38
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Path of b2,b3
for b6 being Path of b3,b4 holds
rng (b5 + b6) = (rng b5) \/ rng b6;
:: JORDAN:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b3,b4
for b8 being Path of b4,b5
st b2,b3 are_connected & b3,b4 are_connected & b4,b5 are_connected
holds rng ((b6 + b7) + b8) = ((rng b6) \/ rng b7) \/ rng b8;
:: JORDAN:th 40
theorem
for b1 being non empty TopSpace-like arcwise_connected TopStruct
for b2, b3, b4, b5 being Element of the carrier of b1
for b6 being Path of b2,b3
for b7 being Path of b3,b4
for b8 being Path of b4,b5 holds
rng ((b6 + b7) + b8) = ((rng b6) \/ rng b7) \/ rng b8;
:: JORDAN:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of the carrier of b1 holds
I[01] --> b2 is Path of b2,b2;
:: JORDAN:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of bool the carrier of TOP-REAL b1
st b4 is_an_arc_of b2,b3
holds ex b5 being Path of b2,b3 st
ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | b4 st
rng b6 = b4 & b5 = b6;
:: JORDAN:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1 holds
ex b4 being Path of b2,b3 st
ex b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | LSeg(b2,b3) st
rng b5 = LSeg(b2,b3) & b4 = b5;
:: JORDAN:th 44
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 <> b2 & b4 <> b3 & b5 <> b2 & b5 <> b3
holds ex b6 being Path of b4,b5 st
rng b6 c= b1 & rng b6 misses {b2,b3};
:: JORDAN:th 45
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds [.b1,b2,b3,b4.] c= closed_inside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:th 46
theorem
for b1, b2, b3, b4 being real set holds
inside_of_rectangle(b1,b2,b3,b4) c= closed_inside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:th 47
theorem
for b1, b2, b3, b4 being real set holds
closed_inside_of_rectangle(b1,b2,b3,b4) = (outside_of_rectangle(b1,b2,b3,b4)) `;
:: JORDAN:funcreg 14
registration
let a1, a2, a3, a4 be real set;
cluster closed_inside_of_rectangle(a1,a2,a3,a4) -> closed;
end;
:: JORDAN:th 48
theorem
for b1, b2, b3, b4 being real set holds
closed_inside_of_rectangle(b1,b2,b3,b4) misses outside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:th 49
theorem
for b1, b2, b3, b4 being real set holds
(closed_inside_of_rectangle(b1,b2,b3,b4)) /\ inside_of_rectangle(b1,b2,b3,b4) = inside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:th 50
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds Int closed_inside_of_rectangle(b1,b2,b3,b4) = inside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:th 51
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds (closed_inside_of_rectangle(b1,b2,b3,b4)) \ inside_of_rectangle(b1,b2,b3,b4) = [.b1,b2,b3,b4.];
:: JORDAN:th 52
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b3 < b4
holds Fr closed_inside_of_rectangle(b1,b2,b3,b4) = [.b1,b2,b3,b4.];
:: JORDAN:th 53
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds W-bound closed_inside_of_rectangle(b1,b2,b3,b4) = b1;
:: JORDAN:th 54
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds S-bound closed_inside_of_rectangle(b1,b2,b3,b4) = b3;
:: JORDAN:th 55
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds E-bound closed_inside_of_rectangle(b1,b2,b3,b4) = b2;
:: JORDAN:th 56
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4
holds N-bound closed_inside_of_rectangle(b1,b2,b3,b4) = b4;
:: JORDAN:th 57
theorem
for b1, b2, b3, b4 being real set
for b5, b6 being Element of the carrier of TOP-REAL 2
for b7 being Element of bool the carrier of TOP-REAL 2
st b1 < b2 & b3 < b4 & b5 in closed_inside_of_rectangle(b1,b2,b3,b4) & not b6 in closed_inside_of_rectangle(b1,b2,b3,b4) & b7 is_an_arc_of b5,b6
holds Segment(b7,b5,b6,b5,First_Point(b7,b5,b6,[.b1,b2,b3,b4.])) c= closed_inside_of_rectangle(b1,b2,b3,b4);
:: JORDAN:funcnot 1 => JORDAN:func 1
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of [:a1,a2:];
redefine func a3 `1 -> Element of the carrier of a1;
end;
:: JORDAN:funcnot 2 => JORDAN:func 2
definition
let a1, a2 be non empty TopSpace-like TopStruct;
let a3 be Element of the carrier of [:a1,a2:];
redefine func a3 `2 -> Element of the carrier of a2;
end;
:: JORDAN:funcnot 3 => JORDAN:func 3
definition
let a1 be Element of the carrier of TOP-REAL 2;
func diffX2_1 A1 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `2 `1 - (a1 `1);
end;
:: JORDAN:def 1
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,TOP-REAL 2:],REAL holds
b2 = diffX2_1 b1
iff
for b3 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b2 . b3 = b3 `2 `1 - (b1 `1);
:: JORDAN:funcnot 4 => JORDAN:func 4
definition
let a1 be Element of the carrier of TOP-REAL 2;
func diffX2_2 A1 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `2 `2 - (a1 `2);
end;
:: JORDAN:def 2
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,TOP-REAL 2:],REAL holds
b2 = diffX2_2 b1
iff
for b3 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b2 . b3 = b3 `2 `2 - (b1 `2);
:: JORDAN:funcnot 5 => JORDAN:func 5
definition
func diffX1_X2_1 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `1 `1 - (b1 `2 `1);
end;
:: JORDAN:def 3
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL holds
b1 = diffX1_X2_1
iff
for b2 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b1 . b2 = b2 `1 `1 - (b2 `2 `1);
:: JORDAN:funcnot 6 => JORDAN:func 6
definition
func diffX1_X2_2 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `1 `2 - (b1 `2 `2);
end;
:: JORDAN:def 4
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL holds
b1 = diffX1_X2_2
iff
for b2 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b1 . b2 = b2 `1 `2 - (b2 `2 `2);
:: JORDAN:funcnot 7 => JORDAN:func 7
definition
func Proj2_1 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `2 `1;
end;
:: JORDAN:def 5
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL holds
b1 = Proj2_1
iff
for b2 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b1 . b2 = b2 `2 `1;
:: JORDAN:funcnot 8 => JORDAN:func 8
definition
func Proj2_2 -> Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL means
for b1 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
it . b1 = b1 `2 `2;
end;
:: JORDAN:def 6
theorem
for b1 being Function-like quasi_total Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],REAL holds
b1 = Proj2_2
iff
for b2 being Element of the carrier of [:TOP-REAL 2,TOP-REAL 2:] holds
b1 . b2 = b2 `2 `2;
:: JORDAN:th 58
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
diffX2_1 b1 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:th 59
theorem
for b1 being Element of the carrier of TOP-REAL 2 holds
diffX2_2 b1 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:th 60
theorem
diffX1_X2_1 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:th 61
theorem
diffX1_X2_2 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:th 62
theorem
Proj2_1 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:th 63
theorem
Proj2_2 is Function-like quasi_total continuous Relation of the carrier of [:TOP-REAL 2,TOP-REAL 2:],the carrier of R^1;
:: JORDAN:funcreg 15
registration
let a1 be Element of the carrier of TOP-REAL 2;
cluster diffX2_1 a1 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcreg 16
registration
let a1 be Element of the carrier of TOP-REAL 2;
cluster diffX2_2 a1 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcreg 17
registration
cluster diffX1_X2_1 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcreg 18
registration
cluster diffX1_X2_2 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcreg 19
registration
cluster Proj2_1 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcreg 20
registration
cluster Proj2_2 -> Function-like quasi_total continuous;
end;
:: JORDAN:funcnot 9 => JORDAN:func 9
definition
let a1 be non empty Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
let a4 be positive real set;
assume a3 is Element of the carrier of Tdisk(a2,a4);
func DiskProj(A2,A4,A3) -> Function-like quasi_total Relation of the carrier of (TOP-REAL a1) | ((cl_Ball(a2,a4)) \ {a3}),the carrier of Tcircle(a2,a4) means
for b1 being Element of the carrier of (TOP-REAL a1) | ((cl_Ball(a2,a4)) \ {a3}) holds
ex b2 being Element of the carrier of TOP-REAL a1 st
b1 = b2 & it . b1 = HC(a3,b2,a2,a4);
end;
:: JORDAN:def 7
theorem
for b1 being non empty Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being positive real set
st b3 is Element of the carrier of Tdisk(b2,b4)
for b5 being Function-like quasi_total Relation of the carrier of (TOP-REAL b1) | ((cl_Ball(b2,b4)) \ {b3}),the carrier of Tcircle(b2,b4) holds
b5 = DiskProj(b2,b4,b3)
iff
for b6 being Element of the carrier of (TOP-REAL b1) | ((cl_Ball(b2,b4)) \ {b3}) holds
ex b7 being Element of the carrier of TOP-REAL b1 st
b6 = b7 & b5 . b6 = HC(b3,b7,b2,b4);
:: JORDAN:th 64
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being positive real set
st b2 is Element of the carrier of Tdisk(b1,b3)
holds DiskProj(b1,b3,b2) is continuous((TOP-REAL 2) | ((cl_Ball(b1,b3)) \ {b2}), Tcircle(b1,b3));
:: JORDAN:th 65
theorem
for b1 being non empty Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being positive real set
st b3 in Ball(b2,b4)
holds (DiskProj(b2,b4,b3)) | Sphere(b2,b4) = id Sphere(b2,b4);
:: JORDAN:funcnot 10 => JORDAN:func 10
definition
let a1 be non empty Element of NAT;
let a2, a3 be Element of the carrier of TOP-REAL a1;
let a4 be positive real set;
assume a3 in Ball(a2,a4);
func RotateCircle(A2,A4,A3) -> Function-like quasi_total Relation of the carrier of Tcircle(a2,a4),the carrier of Tcircle(a2,a4) means
for b1 being Element of the carrier of Tcircle(a2,a4) holds
ex b2 being Element of the carrier of TOP-REAL a1 st
b1 = b2 & it . b1 = HC(b2,a3,a2,a4);
end;
:: JORDAN:def 8
theorem
for b1 being non empty Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being positive real set
st b3 in Ball(b2,b4)
for b5 being Function-like quasi_total Relation of the carrier of Tcircle(b2,b4),the carrier of Tcircle(b2,b4) holds
b5 = RotateCircle(b2,b4,b3)
iff
for b6 being Element of the carrier of Tcircle(b2,b4) holds
ex b7 being Element of the carrier of TOP-REAL b1 st
b6 = b7 & b5 . b6 = HC(b7,b3,b2,b4);
:: JORDAN:th 66
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being positive real set
st b2 in Ball(b1,b3)
holds RotateCircle(b1,b3,b2) is continuous(Tcircle(b1,b3), Tcircle(b1,b3));
:: JORDAN:th 67
theorem
for b1 being non empty Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being positive real set
st b3 in Ball(b2,b4)
holds RotateCircle(b2,b4,b3) has_no_fixpoint;
:: JORDAN:th 68
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
st b3 = b2 & b3 is_a_component_of (TOP-REAL 2) | (b1 `) & b4 is_a_component_of (TOP-REAL 2) | (b1 `) & b3 <> b4
holds Cl b2 misses b4;
:: JORDAN:th 69
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
st b2 is_a_component_of (TOP-REAL 2) | (b1 `)
holds ((TOP-REAL 2) | (b1 `)) | b2 is arcwise_connected;
:: JORDAN:th 70
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Homeomorphism of TOP-REAL 2 holds
b2 .: b1 is being_simple_closed_curve;
:: JORDAN:th 71
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds b1 c= closed_inside_of_rectangle(- 1,1,- 3,3);
:: JORDAN:th 72
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds b1 misses LSeg(|[- 1,3]|,|[1,3]|);
:: JORDAN:th 73
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds b1 misses LSeg(|[- 1,- 3]|,|[1,- 3]|);
:: JORDAN:th 74
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds b1 /\ [.- 1,1,- 3,3.] = {|[- 1,0]|,|[1,0]|};
:: JORDAN:th 75
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds W-bound b1 = - 1;
:: JORDAN:th 76
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds E-bound b1 = 1;
:: JORDAN:th 77
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds W-most b1 = {|[- 1,0]|};
:: JORDAN:th 78
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds E-most b1 = {|[1,0]|};
:: JORDAN:th 79
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds W-min b1 = |[- 1,0]| &
W-max b1 = |[- 1,0]|;
:: JORDAN:th 80
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds E-min b1 = |[1,0]| & E-max b1 = |[1,0]|;
:: JORDAN:th 81
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds LSeg(|[0,3]|,UMP b1) is vertical;
:: JORDAN:th 82
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds LSeg(LMP b1,|[0,- 3]|) is vertical;
:: JORDAN:th 83
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b2 &
b1 in b2
holds b1 `2 < 3;
:: JORDAN:th 84
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b2 &
b1 in b2
holds - 3 < b1 `2;
:: JORDAN:th 85
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b2 &
b1 in LSeg(|[0,3]|,UMP b2)
holds (UMP b2) `2 <= b1 `2;
:: JORDAN:th 86
theorem
for b1 being Element of the carrier of TOP-REAL 2
for b2 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b2 &
b1 in LSeg(LMP b2,|[0,- 3]|)
holds b1 `2 <= (LMP b2) `2;
:: JORDAN:th 87
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds LSeg(|[0,3]|,UMP b1) c= north_halfline UMP b1;
:: JORDAN:th 88
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds LSeg(LMP b1,|[0,- 3]|) c= south_halfline LMP b1;
:: JORDAN:th 89
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1 &
b2 is_inside_component_of b1
holds LSeg(|[0,3]|,UMP b1) misses b2;
:: JORDAN:th 90
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1 &
b2 is_inside_component_of b1
holds LSeg(LMP b1,|[0,- 3]|) misses b2;
:: JORDAN:th 91
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds (LSeg(|[0,3]|,UMP b1)) /\ b1 = {UMP b1};
:: JORDAN:th 92
theorem
for b1 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds (LSeg(|[0,- 3]|,LMP b1)) /\ b1 = {LMP b1};
:: JORDAN:th 93
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 is compact(TOP-REAL 2) &
|[- 1,0]|,|[1,0]| realize-max-dist-in b1 &
b2 is_inside_component_of b1
holds b2 c= closed_inside_of_rectangle(- 1,1,- 3,3);
:: JORDAN:th 94
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
holds LSeg(|[0,3]|,|[0,- 3]|) meets b1;
:: JORDAN:th 95
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
for b2, b3 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st b2 is_an_arc_of |[- 1,0]|,|[1,0]| &
b3 is_an_arc_of |[- 1,0]|,|[1,0]| &
b1 = b2 \/ b3 &
b2 /\ b3 = {|[- 1,0]|,|[1,0]|} &
UMP b1 in b2 &
LMP b1 in b3 &
W-bound b1 = W-bound b2 &
E-bound b1 = E-bound b2
for b4 being Element of bool the carrier of TOP-REAL 2
st b4 = Component_of Down((1 / 2) * ((UMP ((LSeg(LMP b2,|[0,- 3]|)) /\ b3)) + LMP b2),b1 `)
holds b4 is_inside_component_of b1 &
(for b5 being Element of bool the carrier of TOP-REAL 2
st b5 is_inside_component_of b1
holds b5 = b4);
:: JORDAN:th 96
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
st |[- 1,0]|,|[1,0]| realize-max-dist-in b1
for b2, b3 being compact with_the_max_arc Element of bool the carrier of TOP-REAL 2
st b2 is_an_arc_of |[- 1,0]|,|[1,0]| &
b3 is_an_arc_of |[- 1,0]|,|[1,0]| &
b1 = b2 \/ b3 &
b2 /\ b3 = {|[- 1,0]|,|[1,0]|} &
UMP b1 in b2 &
LMP b1 in b3 &
W-bound b1 = W-bound b2 &
E-bound b1 = E-bound b2
holds BDD b1 = Component_of Down((1 / 2) * ((UMP ((LSeg(LMP b2,|[0,- 3]|)) /\ b3)) + LMP b2),b1 `);
:: JORDAN:funcreg 21
registration
let a1 be being_simple_closed_curve Element of bool the carrier of TOP-REAL 2;
cluster BDD a1 -> non empty;
end;
:: JORDAN:th 97
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of bool the carrier of TOP-REAL 2
for b3 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
st b3 = b2 & b3 is_a_component_of (TOP-REAL 2) | (b1 `)
holds b1 = Fr b2;
:: JORDAN:th 98
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
ex b2, b3 being Element of bool the carrier of TOP-REAL 2 st
b1 ` = b2 \/ b3 &
b2 misses b3 &
(Cl b2) \ b2 = (Cl b3) \ b3 &
(for b4, b5 being Element of bool the carrier of (TOP-REAL 2) | (b1 `)
st b4 = b2 & b5 = b3
holds b4 is_a_component_of (TOP-REAL 2) | (b1 `) & b5 is_a_component_of (TOP-REAL 2) | (b1 `));
:: JORDAN:th 99
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
b1 is Jordan;