:: On Outside Fashoda Meet Theorem
:: by Yatsuka Nakamura
::
:: Received July 16, 2001
:: Copyright (c) 2001-2021 Association of Mizar Users

theorem Th1: :: JGRAPH_2:1

for T, T1, T2, S being non empty TopSpace
for f being Function of T1,S
for g being Function of T2,S
for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being object st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )

proof end;

theorem Th2: :: JGRAPH_2:2

for n being Element of NAT
for q2 being Point of (Euclid n)
for q being Point of (TOP-REAL n)
for r being Real st q = q2 holds
Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }

proof end;

theorem Th3: :: JGRAPH_2:3

( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54;

theorem Th4: :: JGRAPH_2:4

1.REAL 2 = <*1,1*> by FINSEQ_2:61;

theorem Th5: :: JGRAPH_2:5

( (1.REAL 2) `1 = 1 & (1.REAL 2) `2 = 1 ) by Th4, EUCLID:52;

theorem Th6: :: JGRAPH_2:6

( dom proj1 = the carrier of (TOP-REAL 2) & dom proj1 = REAL 2 )

proof end;

theorem Th7: :: JGRAPH_2:7

( dom proj2 = the carrier of (TOP-REAL 2) & dom proj2 = REAL 2 )

proof end;

theorem Th8: :: JGRAPH_2:8

for p being Point of (TOP-REAL 2) holds p = |[(proj1 . p),(proj2 . p)]|

proof end;

theorem Th9: :: JGRAPH_2:9

for B being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} holds
( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} )

proof end;

theorem Th10: :: JGRAPH_2:10

for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )

proof end;

theorem Th11: :: JGRAPH_2:11

for p being Point of (TOP-REAL 2)
for G being Subset of (TOP-REAL 2) st G is open & p in G holds
ex r being Real st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G )

proof end;

theorem Th12: :: JGRAPH_2:12

for X, Y, Z being non empty TopSpace
for B being Subset of Y
for C being Subset of Z
for f being Function of X,Y
for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )

proof end;

definition

func Out_In_Sq -> Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) means :Def1: :: JGRAPH_2:def 1
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies it . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or it . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) );

existence
ex b₁ being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b₁ . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b₁ . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) )

proof end;

uniqueness
for b₁, b₂ being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b₁ . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b₁ . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) & ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b₂ . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b₂ . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines Out_In_Sq JGRAPH_2:def 1 :
for b₁ being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) holds
( b₁ = Out_In_Sq iff for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b₁ . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b₁ . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) );

theorem Th13: :: JGRAPH_2:13

for p being Point of (TOP-REAL 2) holds
( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) )

proof end;

theorem Th14: :: JGRAPH_2:14

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

proof end;

theorem Th15: :: JGRAPH_2:15

for D being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | D) st 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. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)

proof end;

theorem Th16: :: JGRAPH_2:16

for D being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | D) st 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. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)

proof end;

Lm1: 0. (TOP-REAL 2) = 0.REAL 2
by EUCLID:66;

theorem Th17: :: JGRAPH_2:17

for K0a being set
for D being non empty Subset of (TOP-REAL 2) st K0a = { 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. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )

proof end;

theorem Th18: :: JGRAPH_2:18

for K0a being set
for D being non empty Subset of (TOP-REAL 2) st K0a = { 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. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )

proof end;

theorem Th19: :: JGRAPH_2:19

for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous 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 = r1 + r2 ) & g is continuous )

proof end;

theorem :: JGRAPH_2:20

for X being non empty TopSpace
for a being Real ex g being Function of X,R^1 st
( ( for p being Point of X holds g . p = a ) & g is continuous )

proof end;

theorem Th21: :: JGRAPH_2:21

for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous 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 = r1 - r2 ) & g is continuous )

proof end;

theorem Th22: :: JGRAPH_2:22

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 st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous )

proof end;

theorem Th23: :: JGRAPH_2:23

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

proof end;

theorem Th24: :: JGRAPH_2:24

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

proof end;

theorem Th25: :: JGRAPH_2:25

for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous 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 = r1 * r2 ) & g is continuous )

proof end;

theorem Th26: :: JGRAPH_2:26

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

proof end;

theorem Th27: :: JGRAPH_2:27

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 st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )

proof end;

theorem Th28: :: JGRAPH_2:28

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 st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous )

proof end;

theorem Th29: :: JGRAPH_2:29

for K0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) holds
f is continuous

proof end;

theorem Th30: :: JGRAPH_2:30

for K0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) holds
f is continuous

proof end;

theorem Th31: :: JGRAPH_2:31

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 = 1 / (p `1) ) & ( 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 Th32: :: JGRAPH_2:32

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 = 1 / (p `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 Th33: :: JGRAPH_2:33

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) / (p `1)) / (p `1) ) & ( 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 Th34: :: JGRAPH_2:34

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) / (p `2)) / (p `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 Th35: :: JGRAPH_2:35

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0)
for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being Real st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ) holds
f is continuous

proof end;

theorem Th36: :: JGRAPH_2:36

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-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. (TOP-REAL 2) ) } holds
f is continuous

proof end;

theorem Th37: :: JGRAPH_2:37

for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-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. (TOP-REAL 2) ) } holds
f is continuous

proof end;

scheme :: JGRAPH_2:sch 1
TopSubset{ P₁[ set ] } :

{ p where p is Point of (TOP-REAL 2) : P₁[p] } is Subset of (TOP-REAL 2)

proof end;

scheme :: JGRAPH_2:sch 2
TopCompl{ P₁[ set ], F₁() -> Subset of (TOP-REAL 2) } :

F₁() ` = { p where p is Point of (TOP-REAL 2) : P₁[p] }

provided

A1: F₁() = { p where p is Point of (TOP-REAL 2) : P₁[p] }

proof end;

Lm2: now :: thesis: for p01, p02, px1, px2 being Real st (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) holds
(p01 - p02) / 2 <= px1 - px2

let p01, p02, px1, px2 be Real; :: thesis: ( (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) implies (p01 - p02) / 2 <= px1 - px2 )
set r0 = (p01 - p02) / 4;
assume (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) ; :: thesis: (p01 - p02) / 2 <= px1 - px2
then (p01 - p02) - (px1 - px2) <= ((p01 - p02) / 4) + ((p01 - p02) / 4) ;
then p01 - p02 <= (px1 - px2) + (((p01 - p02) / 4) + ((p01 - p02) / 4)) by XREAL_1:20;
then (p01 - p02) - ((p01 - p02) / 2) <= px1 - px2 by XREAL_1:20;
hence (p01 - p02) / 2 <= px1 - px2 ; :: thesis: verum

end;

scheme :: JGRAPH_2:sch 3
ClosedSubset{ F₁( Point of (TOP-REAL 2)) -> Real, F₂( Point of (TOP-REAL 2)) -> Real } :

{ p where p is Point of (TOP-REAL 2) : F₁(p) <= F₂(p) } is closed Subset of (TOP-REAL 2)

provided

A1: for p, q being Point of (TOP-REAL 2) holds
( F₁((p - q)) = F₁(p) - F₁(q) & F₂((p - q)) = F₂(p) - F₂(q) ) and
A2: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (F₁((p - q)) ^2) + (F₂((p - q)) ^2)

proof end;

deffunc H₁( Point of (TOP-REAL 2)) -> object = $1 `1 ;

deffunc H₂( Point of (TOP-REAL 2)) -> object = $1 `2 ;

Lm3: for p, q being Point of (TOP-REAL 2) holds
( H₁(p - q) = H₁(p) - H₁(q) & H₂(p - q) = H₂(p) - H₂(q) )
by TOPREAL3:3;

Lm4: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₁(p - q) ^2) + (H₂(p - q) ^2)
by JGRAPH_1:29;

Lm5: { p7 where p7 is Point of (TOP-REAL 2) : H₁(p7) <= H₂(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm3, Lm4);

Lm6: for p, q being Point of (TOP-REAL 2) holds
( H₂(p - q) = H₂(p) - H₂(q) & H₁(p - q) = H₁(p) - H₁(q) )
by TOPREAL3:3;

Lm7: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₂(p - q) ^2) + (H₁(p - q) ^2)
by JGRAPH_1:29;

Lm8: { p7 where p7 is Point of (TOP-REAL 2) : H₂(p7) <= H₁(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm6, Lm7);

deffunc H₃( Point of (TOP-REAL 2)) -> object = - ($1 `1);

deffunc H₄( Point of (TOP-REAL 2)) -> object = - ($1 `2);

Lm9: now :: thesis: for p, q being Point of (TOP-REAL 2) holds
( H₃(p - q) = H₃(p) - H₃(q) & H₂(p - q) = H₂(p) - H₂(q) )

let p, q be Point of (TOP-REAL 2); :: thesis: ( H₃(p - q) = H₃(p) - H₃(q) & H₂(p - q) = H₂(p) - H₂(q) )
thus H₃(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3
.= H₃(p) - H₃(q) ; :: thesis: H₂(p - q) = H₂(p) - H₂(q)
thus H₂(p - q) = H₂(p) - H₂(q) by TOPREAL3:3; :: thesis: verum

end;

Lm10: now :: thesis: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₃(p - q) ^2) + (H₂(p - q) ^2)

let p, q be Point of (TOP-REAL 2); :: thesis: |.(p - q).| ^2 = (H₃(p - q) ^2) + (H₂(p - q) ^2)
H₃(p - q) ^2 = H₁(p - q) ^2 ;
hence |.(p - q).| ^2 = (H₃(p - q) ^2) + (H₂(p - q) ^2) by JGRAPH_1:29; :: thesis: verum

end;

Lm11: { p7 where p7 is Point of (TOP-REAL 2) : H₃(p7) <= H₂(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm9, Lm10);

Lm12: now :: thesis: for p, q being Point of (TOP-REAL 2) holds
( H₂(p - q) = H₂(p) - H₂(q) & H₃(p - q) = H₃(p) - H₃(q) )

let p, q be Point of (TOP-REAL 2); :: thesis: ( H₂(p - q) = H₂(p) - H₂(q) & H₃(p - q) = H₃(p) - H₃(q) )
thus H₂(p - q) = H₂(p) - H₂(q) by TOPREAL3:3; :: thesis: H₃(p - q) = H₃(p) - H₃(q)
thus H₃(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3
.= H₃(p) - H₃(q) ; :: thesis: verum

end;

Lm13: now :: thesis: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₂(p - q) ^2) + (H₃(p - q) ^2)

let p, q be Point of (TOP-REAL 2); :: thesis: |.(p - q).| ^2 = (H₂(p - q) ^2) + (H₃(p - q) ^2)
(- ((p - q) `1)) ^2 = ((p - q) `1) ^2 ;
hence |.(p - q).| ^2 = (H₂(p - q) ^2) + (H₃(p - q) ^2) by JGRAPH_1:29; :: thesis: verum

end;

Lm14: { p7 where p7 is Point of (TOP-REAL 2) : H₂(p7) <= H₃(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm12, Lm13);

Lm15: now :: thesis: for p, q being Point of (TOP-REAL 2) holds
( H₄(p - q) = H₄(p) - H₄(q) & H₁(p - q) = H₁(p) - H₁(q) )

let p, q be Point of (TOP-REAL 2); :: thesis: ( H₄(p - q) = H₄(p) - H₄(q) & H₁(p - q) = H₁(p) - H₁(q) )
thus H₄(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3
.= H₄(p) - H₄(q) ; :: thesis: H₁(p - q) = H₁(p) - H₁(q)
thus H₁(p - q) = H₁(p) - H₁(q) by TOPREAL3:3; :: thesis: verum

end;

Lm16: now :: thesis: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₄(p - q) ^2) + (H₁(p - q) ^2)

let p, q be Point of (TOP-REAL 2); :: thesis: |.(p - q).| ^2 = (H₄(p - q) ^2) + (H₁(p - q) ^2)
(- ((p - q) `2)) ^2 = ((p - q) `2) ^2 ;
hence |.(p - q).| ^2 = (H₄(p - q) ^2) + (H₁(p - q) ^2) by JGRAPH_1:29; :: thesis: verum

end;

Lm17: { p7 where p7 is Point of (TOP-REAL 2) : H₄(p7) <= H₁(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm15, Lm16);

Lm18: now :: thesis: for p, q being Point of (TOP-REAL 2) holds
( H₁(p - q) = H₁(p) - H₁(q) & H₄(p - q) = H₄(p) - H₄(q) )

let p, q be Point of (TOP-REAL 2); :: thesis: ( H₁(p - q) = H₁(p) - H₁(q) & H₄(p - q) = H₄(p) - H₄(q) )
thus H₁(p - q) = H₁(p) - H₁(q) by TOPREAL3:3; :: thesis: H₄(p - q) = H₄(p) - H₄(q)
thus H₄(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3
.= H₄(p) - H₄(q) ; :: thesis: verum

end;

Lm19: now :: thesis: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H₁(p - q) ^2) + (H₄(p - q) ^2)

let p, q be Point of (TOP-REAL 2); :: thesis: |.(p - q).| ^2 = (H₁(p - q) ^2) + (H₄(p - q) ^2)
H₄(p - q) ^2 = H₂(p - q) ^2 ;
hence |.(p - q).| ^2 = (H₁(p - q) ^2) + (H₄(p - q) ^2) by JGRAPH_1:29; :: thesis: verum

end;

Lm20: { p7 where p7 is Point of (TOP-REAL 2) : H₁(p7) <= H₄(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch 3(Lm18, Lm19);

theorem Th38: :: JGRAPH_2:38

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 = Out_In_Sq | K0 & B0 = NonZero (TOP-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. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th39: :: JGRAPH_2:39

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 = Out_In_Sq | K0 & B0 = NonZero (TOP-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. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )

proof end;

theorem Th40: :: JGRAPH_2:40

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

proof end;

theorem Th41: :: JGRAPH_2:41

for B, K0, Kb being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & 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
ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st
( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds
f . s = s ) )

proof end;

theorem Th42: :: JGRAPH_2:42

for f, g being Function of I[01],(TOP-REAL 2)
for K0 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 & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds
rng f meets rng g

proof end;

theorem Th43: :: JGRAPH_2:43

for A, B, C, D being Real
for f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds
f is continuous

proof end;

definition

let A, B, C, D be Real;

func AffineMap (A,B,C,D) -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def2: :: JGRAPH_2:def 2
for t being Point of (TOP-REAL 2) holds it . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|;

existence
ex b₁ being Function of (TOP-REAL 2),(TOP-REAL 2) st
for t being Point of (TOP-REAL 2) holds b₁ . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|

proof end;

uniqueness
for b₁, b₂ being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds b₁ . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) & ( for t being Point of (TOP-REAL 2) holds b₂ . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines AffineMap JGRAPH_2:def 2 :
for A, B, C, D being Real
for b₅ being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( b₅ = AffineMap (A,B,C,D) iff for t being Point of (TOP-REAL 2) holds b₅ . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| );

registration

let a, b, c, d be Real;

cluster AffineMap (a,b,c,d) -> continuous ;

coherence
AffineMap (a,b,c,d) is continuous

proof end;

end;

theorem Th44: :: JGRAPH_2:44

for A, B, C, D being Real st A > 0 & C > 0 holds
AffineMap (A,B,C,D) is one-to-one

proof end;

theorem :: JGRAPH_2:45

for f, g being Function of I[01],(TOP-REAL 2)
for a, b, c, d being Real
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 & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds
rng f meets rng g

proof end;

theorem :: JGRAPH_2:46

( { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } is closed Subset of (TOP-REAL 2) ) by Lm5, Lm8;

theorem :: JGRAPH_2:47

( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } is closed Subset of (TOP-REAL 2) ) by Lm11, Lm14;

theorem :: JGRAPH_2:48

( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2) } is closed Subset of (TOP-REAL 2) ) by Lm17, Lm20;