:: JGRAPH_3 semantic presentation

Lm1: for x being real number holds (x ^2 ) + 1 > 0

proof end;

Lm2: TOP-REAL 2 = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;

Lm3: ( dom proj1 = the carrier of (TOP-REAL 2) & dom proj2 = the carrier of (TOP-REAL 2) )
by FUNCT_2:def 1;

theorem Th1: :: JGRAPH_3:1

for x, y being complex number holds
( not x ^2 = y ^2 or x = y or x = - y )

proof end;

theorem Th2: :: JGRAPH_3:2

for x being complex number holds
( not x ^2 = 1 or x = 1 or x = - 1 )

proof end;

theorem Th3: :: JGRAPH_3:3

for x being real number st 0 <= x & x <= 1 holds
x ^2 <= x

proof end;

Lm4: for a, x being real number st a >= 0 & (x - a) * (x + a) <= 0 holds
( - a <= x & x <= a )
by XREAL_1:95;

theorem :: JGRAPH_3:4

canceled;

theorem Th5: :: JGRAPH_3:5

for x being real number st (x ^2 ) - 1 <= 0 holds
( - 1 <= x & x <= 1 )

proof end;

theorem Th6: :: JGRAPH_3:6

for x, y, z being real number holds
( ( x < y & x < z ) iff x < min y,z )

proof end;

theorem Th7: :: JGRAPH_3:7

for x being real number st 0 < x holds
( x / 3 < x & x / 4 < x ) by XREAL_1:223, XREAL_1:225;

theorem Th8: :: JGRAPH_3:8

for x being real number holds
( ( x >= 1 implies sqrt x >= 1 ) & ( x > 1 implies sqrt x > 1 ) ) by SQUARE_1:83, SQUARE_1:94, SQUARE_1:95;

theorem Th9: :: JGRAPH_3:9

for x, y, z, u being real number st x <= y & z <= u holds
].y,z.[ c= ].x,u.[

proof end;

theorem :: JGRAPH_3:10

for p being Point of (TOP-REAL 2) holds
( |.p.| = sqrt (((p `1 ) ^2 ) + ((p `2 ) ^2 )) & |.p.| ^2 = ((p `1 ) ^2 ) + ((p `2 ) ^2 ) ) by JGRAPH_1:46, JGRAPH_1:47;

theorem :: JGRAPH_3:11

for f being Function
for B, C being set holds (f | B) .: C = f .: (C /\ B)

proof end;

theorem Th12: :: JGRAPH_3:12

for X being TopStruct
for Y being non empty TopStruct
for f being Function of X,Y
for P being Subset of X holds f | P is Function of (X | P),Y

proof end;

theorem Th13: :: JGRAPH_3:13

for X, Y being non empty TopSpace
for p0 being Point of X
for D being non empty Subset of X
for E being non empty Subset of Y
for f being Function of X,Y st D ` = {p0} & E ` = {(f . p0)} & X is_T2 & Y is_T2 & ( for p being Point of (X | D) holds f . p <> f . p0 ) & f | D is continuous Function of (X | D),(Y | E) & ( for V being Subset of Y st f . p0 in V & V is open holds
ex W being Subset of X st
( p0 in W & W is open & f .: W c= V ) ) holds
f is continuous

proof end;

definition

func Sq_Circ -> Function of the carrier of (TOP-REAL 2),the carrier of (TOP-REAL 2) means :Def1: :: JGRAPH_3:def 1
for p being Point of (TOP-REAL 2) holds
( ( p = 0.REAL 2 implies it . p = p ) & ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 implies it . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) & ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) or not p <> 0.REAL 2 or it . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines Sq_Circ JGRAPH_3:def 1 :

theorem Th14: :: JGRAPH_3:14

for p being Point of (TOP-REAL 2) st p <> 0.REAL 2 holds
( ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) implies Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) & ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) or Sq_Circ . p = |[((p `1 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) / (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) )

proof end;

theorem Th15: :: JGRAPH_3:15

for X being non empty TopSpace
for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X ex r being real number st
( f1 . q = r & r >= 0 ) ) 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 = sqrt r1 ) & g is continuous )

proof end;

theorem Th16: :: JGRAPH_3:16

for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( 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) ^2 ) & g is continuous )

proof end;

theorem Th17: :: JGRAPH_3:17

proof end;

theorem Th18: :: JGRAPH_3:18

proof end;

theorem Th19: :: JGRAPH_3:19

proof end;

theorem Th20: :: JGRAPH_3:20

proof end;

Lm5: for K1 being non empty Subset of (TOP-REAL 2)
for q being Point of ((TOP-REAL 2) | K1) holds (proj2 | K1) . q = proj2 . q

proof end;

Lm6: for K1 being non empty Subset of (TOP-REAL 2) holds proj2 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1

proof end;

Lm7: for K1 being non empty Subset of (TOP-REAL 2)
for q being Point of ((TOP-REAL 2) | K1) holds (proj1 | K1) . q = proj1 . q

proof end;

Lm8: for K1 being non empty Subset of (TOP-REAL 2) holds proj1 | K1 is continuous Function of ((TOP-REAL 2) | K1),R^1

proof end;

theorem Th21: :: JGRAPH_3:21

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

proof end;

theorem Th22: :: JGRAPH_3:22

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

proof end;

theorem Th23: :: JGRAPH_3:23

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

proof end;

theorem Th24: :: JGRAPH_3:24

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

proof end;

Lm9: ( ( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1 ) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1 ) ) ) & 1.REAL 2 <> 0.REAL 2 )
by JGRAPH_2:13, REVROT_1:19;

Lm10: for K1 being non empty Subset of (TOP-REAL 2) holds dom (proj2 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1)

proof end;

Lm11: for K1 being non empty Subset of (TOP-REAL 2) holds dom (proj1 * (Sq_Circ | K1)) = the carrier of ((TOP-REAL 2) | K1)

proof end;

Lm12: the carrier of (TOP-REAL 2) \ {(0.REAL 2)} <> {}
by JGRAPH_2:19;

theorem Th25: :: JGRAPH_3:25

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 ) } holds
f is continuous

proof end;

Lm13: ( ( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2 ) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2 ) ) ) & 1.REAL 2 <> 0.REAL 2 )
by JGRAPH_2:13, REVROT_1:19;

theorem Th26: :: JGRAPH_3:26

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Sq_Circ | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) & p <> 0.REAL 2 ) } holds
f is continuous

proof end;

scheme :: JGRAPH_3:sch 1
TopIncl{ P₁[ set ] } :

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

proof end;

scheme :: JGRAPH_3:sch 2
TopInter{ P₁[ set ] } :

{ p where p is Point of (TOP-REAL 2) : ( P₁[p] & p <> 0.REAL 2 ) } = { p7 where p7 is Point of (TOP-REAL 2) : P₁[p7] } /\ (the carrier of (TOP-REAL 2) \ {(0.REAL 2)})

proof end;

theorem Th27: :: JGRAPH_3:27

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 f = Sq_Circ | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th28: :: JGRAPH_3:28

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 f = Sq_Circ | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) & p <> 0.REAL 2 ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th29: :: JGRAPH_3:29

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

proof end;

theorem Th30: :: JGRAPH_3:30

for D being non empty Subset of (TOP-REAL 2) st D = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} holds
D ` = {(0.REAL 2)}

proof end;

theorem Th31: :: JGRAPH_3:31

ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = Sq_Circ & h is continuous )

proof end;

theorem Th32: :: JGRAPH_3:32

Sq_Circ is one-to-one

proof end;

registration

cluster Sq_Circ -> one-to-one ;
coherence by Th32;

end;

theorem Th33: :: JGRAPH_3:33

for Kb, Cb being Subset of (TOP-REAL 2) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } & Cb = { p2 where p2 is Point of (TOP-REAL 2) : |.p2.| = 1 } holds
Sq_Circ .: Kb = Cb

proof end;

theorem Th34: :: JGRAPH_3:34

for P, Kb being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | Kb),((TOP-REAL 2) | P) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } & f is_homeomorphism holds
P is_simple_closed_curve

proof end;

theorem Th35: :: JGRAPH_3:35

for Kb being Subset of (TOP-REAL 2) st Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } holds
( Kb is_simple_closed_curve & Kb is compact )

proof end;

theorem :: JGRAPH_3:36

for Cb being Subset of (TOP-REAL 2) st Cb = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds
Cb is_simple_closed_curve

proof end;

theorem :: JGRAPH_3:37

for K0, C0 being Subset of (TOP-REAL 2) st K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) } & C0 = { p1 where p1 is Point of (TOP-REAL 2) : |.p1.| <= 1 } holds
Sq_Circ " C0 c= K0

proof end;

theorem Th38: :: JGRAPH_3:38

for p being Point of (TOP-REAL 2) holds
( ( p = 0.REAL 2 implies (Sq_Circ " ) . p = 0.REAL 2 ) & ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 implies (Sq_Circ " ) . p = |[((p `1 ) * (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) * (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) & ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) or not p <> 0.REAL 2 or (Sq_Circ " ) . p = |[((p `1 ) * (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) * (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) )

proof end;

theorem Th39: :: JGRAPH_3:39

Sq_Circ " is Function of (TOP-REAL 2),(TOP-REAL 2)

proof end;

theorem Th40: :: JGRAPH_3:40

for p being Point of (TOP-REAL 2) st p <> 0.REAL 2 holds
( ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) implies (Sq_Circ " ) . p = |[((p `1 ) * (sqrt (1 + (((p `1 ) / (p `2 )) ^2 )))),((p `2 ) * (sqrt (1 + (((p `1 ) / (p `2 )) ^2 ))))]| ) & ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) or (Sq_Circ " ) . p = |[((p `1 ) * (sqrt (1 + (((p `2 ) / (p `1 )) ^2 )))),((p `2 ) * (sqrt (1 + (((p `2 ) / (p `1 )) ^2 ))))]| ) )

proof end;

theorem Th41: :: JGRAPH_3:41

proof end;

theorem Th42: :: JGRAPH_3:42

proof end;

theorem Th43: :: JGRAPH_3:43

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

proof end;

theorem Th44: :: JGRAPH_3:44

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

proof end;

theorem Th45: :: JGRAPH_3:45

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

proof end;

theorem Th46: :: JGRAPH_3:46

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

proof end;

Lm14: for K1 being non empty Subset of (TOP-REAL 2) holds proj2 * ((Sq_Circ " ) | K1) is Function of ((TOP-REAL 2) | K1),R^1

proof end;

Lm15: for K1 being non empty Subset of (TOP-REAL 2) holds proj1 * ((Sq_Circ " ) | K1) is Function of ((TOP-REAL 2) | K1),R^1

proof end;

theorem Th47: :: JGRAPH_3:47

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = (Sq_Circ " ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 ) } holds
f is continuous

proof end;

theorem Th48: :: JGRAPH_3:48

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = (Sq_Circ " ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) & p <> 0.REAL 2 ) } holds
f is continuous

proof end;

theorem Th49: :: JGRAPH_3:49

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 f = (Sq_Circ " ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1 ) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1 ) ) ) & p <> 0.REAL 2 ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th50: :: JGRAPH_3:50

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 f = (Sq_Circ " ) | K0 & B0 = the carrier of (TOP-REAL 2) \ {(0.REAL 2)} & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2 ) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2 ) ) ) & p <> 0.REAL 2 ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th51: :: JGRAPH_3:51

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

proof end;

theorem Th52: :: JGRAPH_3:52

ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h = Sq_Circ " & h is continuous )

proof end;

Lm16: Sq_Circ " is one-to-one
by FUNCT_1:62;

theorem :: JGRAPH_3:53

canceled;

theorem Th54: :: JGRAPH_3:54

( Sq_Circ is Function of (TOP-REAL 2),(TOP-REAL 2) & rng Sq_Circ = the carrier of (TOP-REAL 2) & ( for f being Function of (TOP-REAL 2),(TOP-REAL 2) st f = Sq_Circ holds
f is_homeomorphism ) )

proof end;

Lm17: now

let pz2, pz1 be real number ; :: thesis:
assume (((pz2 ^2 ) + (pz1 ^2 )) - 1) * (pz2 ^2 ) <= pz1 ^2 ; :: thesis:
then (((pz2 ^2 ) * (pz2 ^2 )) + ((pz2 ^2 ) * ((pz1 ^2 ) - 1))) - (pz1 ^2 ) <= (pz1 ^2 ) - (pz1 ^2 ) by XREAL_1:11;
hence ((pz2 ^2 ) - 1) * ((pz2 ^2 ) + (pz1 ^2 )) <= 0 ; :: thesis:

end;

Lm18: now

let px1 be real number ; :: thesis:
assume (px1 ^2 ) - 1 = 0 ; :: thesis:
then (px1 - 1) * (px1 + 1) = 0 ;
then ( px1 - 1 = 0 or px1 + 1 = 0 ) by XCMPLX_1:6;
then ( px1 = 0 + 1 or px1 + 1 = 0 ) ;
then ( px1 = 1 or px1 = 0 - 1 ) ;
hence ( px1 = 1 or px1 = - 1 ) ; :: thesis:

end;

theorem :: JGRAPH_3:55

for f, g being Function of I[01] ,(TOP-REAL 2)
for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1 ) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1 ) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1 ) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1 ) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds
rng f meets rng g

proof end;