JGRAPH_4: Fan Homeomorphisms in the Plane

:: Fan Homeomorphisms in the Plane
:: by Yatsuka Nakamura
::
:: Received January 8, 2002
:: Copyright (c) 2002-2012 Association of Mizar Users

begin

Lm1: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
|.p.| > 0

proof end;

theorem Th1: :: JGRAPH_4:1

for X being non empty TopStruct
for g being Function of X,R^1
for B being Subset of X
for a being real number st g is continuous & B = { p where p is Point of X : g /. p > a } holds
B is open

proof end;

theorem Th2: :: JGRAPH_4:2

for X being non empty TopStruct
for g being Function of X,R^1
for B being Subset of X
for a being Real st g is continuous & B = { p where p is Point of X : g /. p < a } holds
B is open

proof end;

theorem Th3: :: JGRAPH_4:3

for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f is continuous & f is one-to-one & rng f = [#] (TOP-REAL 2) & ( for p2 being Point of (TOP-REAL 2) ex K being non empty compact Subset of (TOP-REAL 2) st
( K = f .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & f . p2 in V2 ) ) ) holds
f is being_homeomorphism

proof end;

theorem Th4: :: JGRAPH_4:4

for X being non empty TopSpace
for f1, f2 being Function of X,R^1
for a, b being real number st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = ((r1 / r2) - a) / b ) & g is continuous )

proof end;

theorem Th5: :: JGRAPH_4:5

for X being non empty TopSpace
for f1, f2 being Function of X,R^1
for a, b being Real st f1 is continuous & f2 is continuous & b <> 0 & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being Real st f1 . p = r1 & f2 . p = r2 holds
g . p = r2 * (((r1 / r2) - a) / b) ) & g is continuous )

proof end;

theorem Th6: :: JGRAPH_4:6

for X being non empty TopSpace
for f1 being Function of X,R^1 st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 ^2 ) & g is continuous )

proof end;

theorem Th7: :: JGRAPH_4:7

proof end;

theorem Th8: :: JGRAPH_4:8

proof end;

theorem Th9: :: JGRAPH_4:9

proof end;

theorem Th10: :: JGRAPH_4:10

proof end;

definition

let n be Nat;
deffunc H₁( Point of (TOP-REAL n)) -> Element of REAL = |.$1.|;

func n NormF -> Function of (TOP-REAL n),R^1 means :Def1: :: JGRAPH_4:def 1
for q being Point of (TOP-REAL n) holds it . q = |.q.|;

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines NormF JGRAPH_4:def 1 :

theorem :: JGRAPH_4:11

for n being Nat holds
( dom (n NormF) = the carrier of (TOP-REAL n) & dom (n NormF) = REAL n )

proof end;

theorem Th12: :: JGRAPH_4:12

for n being Nat holds n NormF is continuous

proof end;

registration

let n be Nat;

cluster n NormF -> continuous ;

coherence by Th12;

end;

theorem Th13: :: JGRAPH_4:13

for n being Element of NAT
for K0 being Subset of (TOP-REAL n)
for f being Function of ((TOP-REAL n) | K0),R^1 st ( for p being Point of ((TOP-REAL n) | K0) holds f . p = (n NormF) . p ) holds
f is continuous

proof end;

theorem Th14: :: JGRAPH_4:14

for n being Element of NAT
for p being Point of (Euclid n)
for r being Real
for B being Subset of (TOP-REAL n) st B = cl_Ball (p,r) holds
( B is bounded & B is closed )

proof end;

theorem Th15: :: JGRAPH_4:15

for p being Point of (Euclid 2)
for r being Real
for B being Subset of (TOP-REAL 2) st B = cl_Ball (p,r) holds
B is compact

proof end;

begin

definition

let s be real number ;
let q be Point of (TOP-REAL 2);

func FanW (s,q) -> Point of (TOP-REAL 2) equals :Def2: :: JGRAPH_4:def 2
|.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2)))),((((q `2) / |.q.|) - s) / (1 - s))]| if ( (q `2) / |.q.| >= s & q `1 < 0 )
|.q.| * |[(- (sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2)))),((((q `2) / |.q.|) - s) / (1 + s))]| if ( (q `2) / |.q.| < s & q `1 < 0 )
otherwise q;

correctness ;

end;

:: deftheorem Def2 defines FanW JGRAPH_4:def 2 :

definition

let s be real number ;

func s -FanMorphW -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def3: :: JGRAPH_4:def 3
for q being Point of (TOP-REAL 2) holds it . q = FanW (s,q);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines -FanMorphW JGRAPH_4:def 3 :

theorem Th16: :: JGRAPH_4:16

for q being Point of (TOP-REAL 2)
for sn being real number holds
( ( (q `2) / |.q.| >= sn & q `1 < 0 implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 >= 0 implies (sn -FanMorphW) . q = q ) )

proof end;

theorem Th17: :: JGRAPH_4:17

for q being Point of (TOP-REAL 2)
for sn being Real st (q `2) / |.q.| <= sn & q `1 < 0 holds
(sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|

proof end;

theorem Th18: :: JGRAPH_4:18

for q being Point of (TOP-REAL 2)
for sn being Real st - 1 < sn & sn < 1 holds
( ( (q `2) / |.q.| >= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 <= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphW) . q = |[(|.q.| * (- (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2))))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) )

proof end;

Lm2: for K being non empty Subset of (TOP-REAL 2) holds
( proj1 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj1 | K) . q = proj1 . q ) )

proof end;

Lm3: for K being non empty Subset of (TOP-REAL 2) holds
( proj2 | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds (proj2 | K) . q = proj2 . q ) )

proof end;

Lm4: dom (2 NormF) = the carrier of (TOP-REAL 2)
by FUNCT_2:def 1;

Lm5: for K being non empty Subset of (TOP-REAL 2) holds
( (2 NormF) | K is continuous Function of ((TOP-REAL 2) | K),R^1 & ( for q being Point of ((TOP-REAL 2) | K) holds ((2 NormF) | K) . q = (2 NormF) . q ) )

proof end;

Lm6: for K1 being non empty Subset of (TOP-REAL 2)
for g1 being Function of ((TOP-REAL 2) | K1),R^1 st g1 = (2 NormF) | K1 & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q <> 0. (TOP-REAL 2) ) holds
for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0

proof end;

theorem Th19: :: JGRAPH_4:19

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th20: :: JGRAPH_4:20

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th21: :: JGRAPH_4:21

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 <= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th22: :: JGRAPH_4:22

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (- (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 <= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th23: :: JGRAPH_4:23

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th24: :: JGRAPH_4:24

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

Lm7: for sn being Real
for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 >= sn * |.p7.| } holds
K1 is closed

proof end;

Lm8: for sn being Real
for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 >= sn * |.p7.| } holds
K1 is closed

proof end;

theorem Th25: :: JGRAPH_4:25

for sn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 <= 0 ) } holds
K03 is closed

proof end;

Lm9: for sn being Real
for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= sn * |.p7.| } holds
K1 is closed

proof end;

Lm10: for sn being Real
for K1 being Subset of (TOP-REAL 2) st K1 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= sn * |.p7.| } holds
K1 is closed

proof end;

theorem Th26: :: JGRAPH_4:26

for sn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 <= 0 ) } holds
K03 is closed

proof end;

theorem Th27: :: JGRAPH_4:27

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th28: :: JGRAPH_4:28

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th29: :: JGRAPH_4:29

for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
K0 is closed

proof end;

theorem Th30: :: JGRAPH_4:30

for sn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th31: :: JGRAPH_4:31

for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
K0 is closed

proof end;

theorem Th32: :: JGRAPH_4:32

for sn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphW) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th33: :: JGRAPH_4:33

for sn being Real
for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphW) . p).| = |.p.|

proof end;

theorem Th34: :: JGRAPH_4:34

for sn being Real
for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
(sn -FanMorphW) . x in K0

proof end;

theorem Th35: :: JGRAPH_4:35

for sn being Real
for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(sn -FanMorphW) . x in K0

proof end;

scheme :: JGRAPH_4:sch 1
InclSub{ F₁() -> non empty Subset of (TOP-REAL 2), P₁[ set ] } :

{ p where p is Point of (TOP-REAL 2) : ( P₁[p] & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | F₁())

provided

A1: F₁() = NonZero (TOP-REAL 2)

proof end;

theorem Th36: :: JGRAPH_4:36

for sn being Real
for D being non empty Subset of (TOP-REAL 2) st - 1 < sn & sn < 1 & D ` = {(0. (TOP-REAL 2))} holds
ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (sn -FanMorphW) | D & h is continuous )

proof end;

Lm11: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;

theorem Th37: :: JGRAPH_4:37

for sn being Real st - 1 < sn & sn < 1 holds
ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = sn -FanMorphW & h is continuous )

proof end;

theorem Th38: :: JGRAPH_4:38

for sn being Real st - 1 < sn & sn < 1 holds
sn -FanMorphW is one-to-one

proof end;

theorem Th39: :: JGRAPH_4:39

for sn being Real st - 1 < sn & sn < 1 holds
( sn -FanMorphW is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphW) = the carrier of (TOP-REAL 2) )

proof end;

Lm12: for q4, q, p2 being Point of (TOP-REAL 2)
for O, u, uq being Point of (Euclid 2) st u in cl_Ball (O,(|.p2.| + 1)) & q = uq & q4 = u & O = 0. (TOP-REAL 2) & |.q4.| = |.q.| holds
q in cl_Ball (O,(|.p2.| + 1))

proof end;

theorem Th40: :: JGRAPH_4:40

for sn being Real
for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds
ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphW) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphW) . p2 in V2 ) )

proof end;

theorem :: JGRAPH_4:41

for sn being Real st - 1 < sn & sn < 1 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = sn -FanMorphW & f is being_homeomorphism )

proof end;

Lm13: now :: thesis:

let q be Point of (TOP-REAL 2); :: thesis:
let sn, t be Real; :: thesis:
assume ((- ((t / |.q.|) - sn)) / (1 - sn)) ^2 < 1 ^2 ; :: thesis:
then 1 - (((- ((t / |.q.|) - sn)) / (1 - sn)) ^2) > 0 by XREAL_1:50;
then sqrt (1 - (((- ((t / |.q.|) - sn)) / (1 - sn)) ^2)) > 0 by SQUARE_1:25;
then sqrt (1 - (((- ((t / |.q.|) - sn)) ^2) / ((1 - sn) ^2))) > 0 by XCMPLX_1:76;
then sqrt (1 - ((((t / |.q.|) - sn) ^2) / ((1 - sn) ^2))) > 0 ;
then sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2)) > 0 by XCMPLX_1:76;
hence - (sqrt (1 - ((((t / |.q.|) - sn) / (1 - sn)) ^2))) < - 0 by XREAL_1:24; :: thesis:

end;

theorem Th42: :: JGRAPH_4:42

for sn being Real
for q being Point of (TOP-REAL 2) st sn < 1 & q `1 < 0 & (q `2) / |.q.| >= sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds
( p `1 < 0 & p `2 >= 0 )

proof end;

theorem Th43: :: JGRAPH_4:43

for sn being Real
for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 < 0 & (q `2) / |.q.| < sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds
( p `1 < 0 & p `2 < 0 )

proof end;

theorem Th44: :: JGRAPH_4:44

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 < 0 & (q1 `2) / |.q1.| >= sn & q2 `1 < 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem Th45: :: JGRAPH_4:45

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 < 0 & (q1 `2) / |.q1.| < sn & q2 `1 < 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem :: JGRAPH_4:46

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 < 0 & q2 `1 < 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphW) . q1 & p2 = (sn -FanMorphW) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem :: JGRAPH_4:47

for sn being Real
for q being Point of (TOP-REAL 2) st q `1 < 0 & (q `2) / |.q.| = sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphW) . q holds
( p `1 < 0 & p `2 = 0 )

proof end;

theorem :: JGRAPH_4:48

for sn being real number holds 0. (TOP-REAL 2) = (sn -FanMorphW) . (0. (TOP-REAL 2)) by Th16, JGRAPH_2:3;

begin

definition

let s be real number ;
let q be Point of (TOP-REAL 2);

func FanN (s,q) -> Point of (TOP-REAL 2) equals :Def4: :: JGRAPH_4:def 4
|.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2)))]| if ( (q `1) / |.q.| >= s & q `2 > 0 )
|.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2)))]| if ( (q `1) / |.q.| < s & q `2 > 0 )
otherwise q;

correctness ;

end;

:: deftheorem Def4 defines FanN JGRAPH_4:def 4 :

definition

let c be real number ;

func c -FanMorphN -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def5: :: JGRAPH_4:def 5
for q being Point of (TOP-REAL 2) holds it . q = FanN (c,q);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines -FanMorphN JGRAPH_4:def 5 :

theorem Th49: :: JGRAPH_4:49

for q being Point of (TOP-REAL 2)
for cn being real number holds
( ( (q `1) / |.q.| >= cn & q `2 > 0 implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( q `2 <= 0 implies (cn -FanMorphN) . q = q ) )

proof end;

theorem Th50: :: JGRAPH_4:50

for q being Point of (TOP-REAL 2)
for cn being Real st (q `1) / |.q.| <= cn & q `2 > 0 holds
(cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]|

proof end;

theorem Th51: :: JGRAPH_4:51

for q being Point of (TOP-REAL 2)
for cn being Real st - 1 < cn & cn < 1 holds
( ( (q `1) / |.q.| >= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2))))]| ) & ( (q `1) / |.q.| <= cn & q `2 >= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphN) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2))))]| ) )

proof end;

theorem Th52: :: JGRAPH_4:52

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th53: :: JGRAPH_4:53

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th54: :: JGRAPH_4:54

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th55: :: JGRAPH_4:55

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 >= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th56: :: JGRAPH_4:56

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th57: :: JGRAPH_4:57

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th58: :: JGRAPH_4:58

for cn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 >= 0 ) } holds
K03 is closed

proof end;

theorem Th59: :: JGRAPH_4:59

for cn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 >= 0 ) } holds
K03 is closed

proof end;

theorem Th60: :: JGRAPH_4:60

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th61: :: JGRAPH_4:61

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th62: :: JGRAPH_4:62

for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
K0 is closed

proof end;

theorem Th63: :: JGRAPH_4:63

for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0) st B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
K0 is closed

proof end;

theorem Th64: :: JGRAPH_4:64

for cn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th65: :: JGRAPH_4:65

for cn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphN) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th66: :: JGRAPH_4:66

for cn being Real
for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphN) . p).| = |.p.|

proof end;

theorem Th67: :: JGRAPH_4:67

for cn being Real
for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(cn -FanMorphN) . x in K0

proof end;

theorem Th68: :: JGRAPH_4:68

for cn being Real
for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
(cn -FanMorphN) . x in K0

proof end;

theorem Th69: :: JGRAPH_4:69

for cn being Real
for D being non empty Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & D ` = {(0. (TOP-REAL 2))} holds
ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = (cn -FanMorphN) | D & h is continuous )

proof end;

theorem Th70: :: JGRAPH_4:70

for cn being Real st - 1 < cn & cn < 1 holds
ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = cn -FanMorphN & h is continuous )

proof end;

theorem Th71: :: JGRAPH_4:71

for cn being Real st - 1 < cn & cn < 1 holds
cn -FanMorphN is one-to-one

proof end;

theorem Th72: :: JGRAPH_4:72

for cn being Real st - 1 < cn & cn < 1 holds
( cn -FanMorphN is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphN) = the carrier of (TOP-REAL 2) )

proof end;

theorem Th73: :: JGRAPH_4:73

for cn being Real
for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds
ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (cn -FanMorphN) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphN) . p2 in V2 ) )

proof end;

theorem :: JGRAPH_4:74

for cn being Real st - 1 < cn & cn < 1 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = cn -FanMorphN & f is being_homeomorphism )

proof end;

theorem Th75: :: JGRAPH_4:75

for cn being Real
for q being Point of (TOP-REAL 2) st cn < 1 & q `2 > 0 & (q `1) / |.q.| >= cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 > 0 & p `1 >= 0 )

proof end;

theorem Th76: :: JGRAPH_4:76

for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 > 0 & (q `1) / |.q.| < cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 > 0 & p `1 < 0 )

proof end;

theorem Th77: :: JGRAPH_4:77

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 > 0 & (q1 `1) / |.q1.| >= cn & q2 `2 > 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem Th78: :: JGRAPH_4:78

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 > 0 & (q1 `1) / |.q1.| < cn & q2 `2 > 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem :: JGRAPH_4:79

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 > 0 & q2 `2 > 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem :: JGRAPH_4:80

for cn being Real
for q being Point of (TOP-REAL 2) st q `2 > 0 & (q `1) / |.q.| = cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 > 0 & p `1 = 0 )

proof end;

theorem :: JGRAPH_4:81

for cn being real number holds 0. (TOP-REAL 2) = (cn -FanMorphN) . (0. (TOP-REAL 2)) by Th49, JGRAPH_2:3;

begin

definition

let s be real number ;
let q be Point of (TOP-REAL 2);

func FanE (s,q) -> Point of (TOP-REAL 2) equals :Def6: :: JGRAPH_4:def 6
|.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 - s)) ^2))),((((q `2) / |.q.|) - s) / (1 - s))]| if ( (q `2) / |.q.| >= s & q `1 > 0 )
|.q.| * |[(sqrt (1 - (((((q `2) / |.q.|) - s) / (1 + s)) ^2))),((((q `2) / |.q.|) - s) / (1 + s))]| if ( (q `2) / |.q.| < s & q `1 > 0 )
otherwise q;

correctness ;

end;

:: deftheorem Def6 defines FanE JGRAPH_4:def 6 :

definition

let s be real number ;

func s -FanMorphE -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def7: :: JGRAPH_4:def 7
for q being Point of (TOP-REAL 2) holds it . q = FanE (s,q);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines -FanMorphE JGRAPH_4:def 7 :

theorem Th82: :: JGRAPH_4:82

for q being Point of (TOP-REAL 2)
for sn being real number holds
( ( (q `2) / |.q.| >= sn & q `1 > 0 implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( q `1 <= 0 implies (sn -FanMorphE) . q = q ) )

proof end;

theorem Th83: :: JGRAPH_4:83

for q being Point of (TOP-REAL 2)
for sn being Real st (q `2) / |.q.| <= sn & q `1 > 0 holds
(sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]|

proof end;

theorem Th84: :: JGRAPH_4:84

for q being Point of (TOP-REAL 2)
for sn being Real st - 1 < sn & sn < 1 holds
( ( (q `2) / |.q.| >= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 - sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 - sn)))]| ) & ( (q `2) / |.q.| <= sn & q `1 >= 0 & q <> 0. (TOP-REAL 2) implies (sn -FanMorphE) . q = |[(|.q.| * (sqrt (1 - (((((q `2) / |.q.|) - sn) / (1 + sn)) ^2)))),(|.q.| * ((((q `2) / |.q.|) - sn) / (1 + sn)))]| ) )

proof end;

theorem Th85: :: JGRAPH_4:85

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 - sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th86: :: JGRAPH_4:86

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `2) / |.p.|) - sn) / (1 + sn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th87: :: JGRAPH_4:87

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st sn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 - sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 >= 0 & (q `2) / |.q.| >= sn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th88: :: JGRAPH_4:88

for sn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < sn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (sqrt (1 - (((((p `2) / |.p.|) - sn) / (1 + sn)) ^2))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `1 >= 0 & (q `2) / |.q.| <= sn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th89: :: JGRAPH_4:89

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| >= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th90: :: JGRAPH_4:90

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `1 >= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `2) / |.p.| <= sn & p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th91: :: JGRAPH_4:91

for sn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= sn * |.p.| & p `1 >= 0 ) } holds
K03 is closed

proof end;

theorem Th92: :: JGRAPH_4:92

for sn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= sn * |.p.| & p `1 >= 0 ) } holds
K03 is closed

proof end;

theorem Th93: :: JGRAPH_4:93

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th94: :: JGRAPH_4:94

for sn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th95: :: JGRAPH_4:95

for sn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th96: :: JGRAPH_4:96

for sn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < sn & sn < 1 & f = (sn -FanMorphE) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th97: :: JGRAPH_4:97

for sn being Real
for p being Point of (TOP-REAL 2) holds |.((sn -FanMorphE) . p).| = |.p.|

proof end;

theorem Th98: :: JGRAPH_4:98

for sn being Real
for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(sn -FanMorphE) . x in K0

proof end;

theorem Th99: :: JGRAPH_4:99

for sn being Real
for x, K0 being set st - 1 < sn & sn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
(sn -FanMorphE) . x in K0

proof end;

theorem Th100: :: JGRAPH_4:100

proof end;

theorem Th101: :: JGRAPH_4:101

for sn being Real st - 1 < sn & sn < 1 holds
ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = sn -FanMorphE & h is continuous )

proof end;

theorem Th102: :: JGRAPH_4:102

for sn being Real st - 1 < sn & sn < 1 holds
sn -FanMorphE is one-to-one

proof end;

theorem Th103: :: JGRAPH_4:103

for sn being Real st - 1 < sn & sn < 1 holds
( sn -FanMorphE is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (sn -FanMorphE) = the carrier of (TOP-REAL 2) )

proof end;

theorem Th104: :: JGRAPH_4:104

for sn being Real
for p2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 holds
ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (sn -FanMorphE) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (sn -FanMorphE) . p2 in V2 ) )

proof end;

theorem :: JGRAPH_4:105

for sn being Real st - 1 < sn & sn < 1 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = sn -FanMorphE & f is being_homeomorphism )

proof end;

theorem Th106: :: JGRAPH_4:106

for sn being Real
for q being Point of (TOP-REAL 2) st sn < 1 & q `1 > 0 & (q `2) / |.q.| >= sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 > 0 & p `2 >= 0 )

proof end;

theorem Th107: :: JGRAPH_4:107

for sn being Real
for q being Point of (TOP-REAL 2) st - 1 < sn & q `1 > 0 & (q `2) / |.q.| < sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 > 0 & p `2 < 0 )

proof end;

theorem Th108: :: JGRAPH_4:108

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st sn < 1 & q1 `1 > 0 & (q1 `2) / |.q1.| >= sn & q2 `1 > 0 & (q2 `2) / |.q2.| >= sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem Th109: :: JGRAPH_4:109

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & q1 `1 > 0 & (q1 `2) / |.q1.| < sn & q2 `1 > 0 & (q2 `2) / |.q2.| < sn & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem :: JGRAPH_4:110

for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 > 0 & q2 `1 > 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 holds
(p1 `2) / |.p1.| < (p2 `2) / |.p2.|

proof end;

theorem :: JGRAPH_4:111

for sn being Real
for q being Point of (TOP-REAL 2) st q `1 > 0 & (q `2) / |.q.| = sn holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 > 0 & p `2 = 0 )

proof end;

theorem :: JGRAPH_4:112

for sn being real number holds 0. (TOP-REAL 2) = (sn -FanMorphE) . (0. (TOP-REAL 2)) by Th82, JGRAPH_2:3;

begin

definition

let s be real number ;
let q be Point of (TOP-REAL 2);

func FanS (s,q) -> Point of (TOP-REAL 2) equals :Def8: :: JGRAPH_4:def 8
|.q.| * |[((((q `1) / |.q.|) - s) / (1 - s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 - s)) ^2))))]| if ( (q `1) / |.q.| >= s & q `2 < 0 )
|.q.| * |[((((q `1) / |.q.|) - s) / (1 + s)),(- (sqrt (1 - (((((q `1) / |.q.|) - s) / (1 + s)) ^2))))]| if ( (q `1) / |.q.| < s & q `2 < 0 )
otherwise q;

correctness ;

end;

:: deftheorem Def8 defines FanS JGRAPH_4:def 8 :

definition

let c be real number ;

func c -FanMorphS -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def9: :: JGRAPH_4:def 9
for q being Point of (TOP-REAL 2) holds it . q = FanS (c,q);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines -FanMorphS JGRAPH_4:def 9 :

theorem Th113: :: JGRAPH_4:113

for q being Point of (TOP-REAL 2)
for cn being real number holds
( ( (q `1) / |.q.| >= cn & q `2 < 0 implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( q `2 >= 0 implies (cn -FanMorphS) . q = q ) )

proof end;

theorem Th114: :: JGRAPH_4:114

for q being Point of (TOP-REAL 2)
for cn being Real st (q `1) / |.q.| <= cn & q `2 < 0 holds
(cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]|

proof end;

theorem Th115: :: JGRAPH_4:115

for q being Point of (TOP-REAL 2)
for cn being Real st - 1 < cn & cn < 1 holds
( ( (q `1) / |.q.| >= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 - cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 - cn)) ^2)))))]| ) & ( (q `1) / |.q.| <= cn & q `2 <= 0 & q <> 0. (TOP-REAL 2) implies (cn -FanMorphS) . q = |[(|.q.| * ((((q `1) / |.q.|) - cn) / (1 + cn))),(|.q.| * (- (sqrt (1 - (((((q `1) / |.q.|) - cn) / (1 + cn)) ^2)))))]| ) )

proof end;

theorem Th116: :: JGRAPH_4:116

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 - cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th117: :: JGRAPH_4:117

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * ((((p `1) / |.p.|) - cn) / (1 + cn)) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th118: :: JGRAPH_4:118

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st cn < 1 & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 - cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & (q `1) / |.q.| >= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th119: :: JGRAPH_4:119

for cn being Real
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st - 1 < cn & ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = |.p.| * (- (sqrt (1 - (((((p `1) / |.p.|) - cn) / (1 + cn)) ^2)))) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
( q `2 <= 0 & (q `1) / |.q.| <= cn & q <> 0. (TOP-REAL 2) ) ) holds
f is continuous

proof end;

theorem Th120: :: JGRAPH_4:120

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| >= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th121: :: JGRAPH_4:121

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = { q where q is Point of (TOP-REAL 2) : ( q `2 <= 0 & q <> 0. (TOP-REAL 2) ) } & K0 = { p where p is Point of (TOP-REAL 2) : ( (p `1) / |.p.| <= cn & p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th122: :: JGRAPH_4:122

for cn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 >= cn * |.p.| & p `2 <= 0 ) } holds
K03 is closed

proof end;

theorem Th123: :: JGRAPH_4:123

for cn being Real
for K03 being Subset of (TOP-REAL 2) st K03 = { p where p is Point of (TOP-REAL 2) : ( p `1 <= cn * |.p.| & p `2 <= 0 ) } holds
K03 is closed

proof end;

theorem Th124: :: JGRAPH_4:124

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th125: :: JGRAPH_4:125

for cn being Real
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th126: :: JGRAPH_4:126

for cn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th127: :: JGRAPH_4:127

for cn being Real
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st - 1 < cn & cn < 1 & f = (cn -FanMorphS) | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th128: :: JGRAPH_4:128

for cn being Real
for p being Point of (TOP-REAL 2) holds |.((cn -FanMorphS) . p).| = |.p.|

proof end;

theorem Th129: :: JGRAPH_4:129

for cn being Real
for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 <= 0 & p <> 0. (TOP-REAL 2) ) } holds
(cn -FanMorphS) . x in K0

proof end;

theorem Th130: :: JGRAPH_4:130

for cn being Real
for x, K0 being set st - 1 < cn & cn < 1 & x in K0 & K0 = { p where p is Point of (TOP-REAL 2) : ( p `2 >= 0 & p <> 0. (TOP-REAL 2) ) } holds
(cn -FanMorphS) . x in K0

proof end;

theorem Th131: :: JGRAPH_4:131

proof end;

theorem Th132: :: JGRAPH_4:132

for cn being Real st - 1 < cn & cn < 1 holds
cn -FanMorphS is continuous

proof end;

theorem Th133: :: JGRAPH_4:133

for cn being Real st - 1 < cn & cn < 1 holds
cn -FanMorphS is one-to-one

proof end;

theorem Th134: :: JGRAPH_4:134

for cn being Real st - 1 < cn & cn < 1 holds
( cn -FanMorphS is Function of (TOP-REAL 2),(TOP-REAL 2) & rng (cn -FanMorphS) = the carrier of (TOP-REAL 2) )

proof end;

theorem Th135: :: JGRAPH_4:135

for cn being Real
for p2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 holds
ex K being non empty compact Subset of (TOP-REAL 2) st
( K = (cn -FanMorphS) .: K & ex V2 being Subset of (TOP-REAL 2) st
( p2 in V2 & V2 is open & V2 c= K & (cn -FanMorphS) . p2 in V2 ) )

proof end;

theorem :: JGRAPH_4:136

for cn being Real st - 1 < cn & cn < 1 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f = cn -FanMorphS & f is being_homeomorphism )

proof end;

theorem Th137: :: JGRAPH_4:137

for cn being Real
for q being Point of (TOP-REAL 2) st cn < 1 & q `2 < 0 & (q `1) / |.q.| >= cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
( p `2 < 0 & p `1 >= 0 )

proof end;

theorem Th138: :: JGRAPH_4:138

for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & q `2 < 0 & (q `1) / |.q.| < cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
( p `2 < 0 & p `1 < 0 )

proof end;

theorem Th139: :: JGRAPH_4:139

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st cn < 1 & q1 `2 < 0 & (q1 `1) / |.q1.| >= cn & q2 `2 < 0 & (q2 `1) / |.q2.| >= cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem Th140: :: JGRAPH_4:140

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & q1 `2 < 0 & (q1 `1) / |.q1.| < cn & q2 `2 < 0 & (q2 `1) / |.q2.| < cn & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem :: JGRAPH_4:141

for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 < 0 & q2 `2 < 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| holds
for p1, p2 being Point of (TOP-REAL 2) st p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 holds
(p1 `1) / |.p1.| < (p2 `1) / |.p2.|

proof end;

theorem :: JGRAPH_4:142

for cn being Real
for q being Point of (TOP-REAL 2) st q `2 < 0 & (q `1) / |.q.| = cn holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
( p `2 < 0 & p `1 = 0 )

proof end;

theorem :: JGRAPH_4:143

for a being real number holds 0. (TOP-REAL 2) = (a -FanMorphS) . (0. (TOP-REAL 2)) by Th113, JGRAPH_2:3;