:: JORDAN_A semantic presentation

theorem Th1: :: JORDAN_A:1

for A, B being non empty finite Subset of REAL holds min (A \/ B) = min (min A),(min B)

proof end;

registration

let T be non empty TopSpace;
cluster non empty compact Element of bool the carrier of T;
existence

proof end;

end;

theorem Th2: :: JORDAN_A:2

for T being non empty TopSpace
for f being continuous RealMap of T
for A being compact Subset of T holds f .: A is compact

proof end;

theorem Th3: :: JORDAN_A:3

for A being compact Subset of REAL
for B being non empty Subset of REAL st B c= A holds
inf B in A

proof end;

theorem :: JORDAN_A:4

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n)
for f being continuous RealMap of [:(TOP-REAL n),(TOP-REAL n):]
for g being RealMap of (TOP-REAL n) st ( for p being Point of (TOP-REAL n) ex G being Subset of REAL st
( G = { (f . p,q) where q is Point of (TOP-REAL n) : q in B } & g . p = inf G ) ) holds
inf (f .: [:A,B:]) = inf (g .: A)

proof end;

theorem :: JORDAN_A:5

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n)
for f being continuous RealMap of [:(TOP-REAL n),(TOP-REAL n):]
for g being RealMap of (TOP-REAL n) st ( for q being Point of (TOP-REAL n) ex G being Subset of REAL st
( G = { (f . p,q) where p is Point of (TOP-REAL n) : p in A } & g . q = inf G ) ) holds
inf (f .: [:A,B:]) = inf (g .: B)

proof end;

theorem Th6: :: JORDAN_A:6

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in Lower_Arc C & q <> W-min C holds
LE E-max C,q,C

proof end;

theorem Th7: :: JORDAN_A:7

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in Upper_Arc C holds
LE q, E-max C,C

proof end;

definition

let n be Nat;
A1: [:the carrier of (TOP-REAL n),the carrier of (TOP-REAL n):] = the carrier of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:def 5;
func Eucl_dist n -> RealMap of [:(TOP-REAL n),(TOP-REAL n):] means :Def1: :: JORDAN_A:def 1
for p, q being Point of (TOP-REAL n) holds it . p,q = |.(p - q).|;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines Eucl_dist JORDAN_A:def 1 :

definition

let T be non empty TopSpace;
let f be RealMap of T;
redefine attr f is continuous means :: JORDAN_A:def 2
for p being Point of T
for N being Neighbourhood of f . p ex V being a_neighborhood of p st f .: V c= N;
compatibility

proof end;

end;

:: deftheorem defines continuous JORDAN_A:def 2 :

registration

let n be Nat;
cluster Eucl_dist n -> continuous ;
coherence

proof end;

end;

theorem Th8: :: JORDAN_A:8

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n) st A misses B holds
dist_min A,B > 0

proof end;

theorem Th9: :: JORDAN_A:9

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p,q,C & LE q, E-max C,C & p <> q holds
Segment p,q,C = Segment (Upper_Arc C),(W-min C),(E-max C),p,q

proof end;

theorem Th10: :: JORDAN_A:10

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st LE E-max C,q,C holds
Segment (E-max C),q,C = Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C),q

proof end;

theorem Th11: :: JORDAN_A:11

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st LE E-max C,q,C holds
Segment q,(W-min C),C = Segment (Lower_Arc C),(E-max C),(W-min C),q,(W-min C)

proof end;

theorem Th12: :: JORDAN_A:12

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p,q,C & LE E-max C,p,C holds
Segment p,q,C = Segment (Lower_Arc C),(E-max C),(W-min C),p,q

proof end;

theorem Th13: :: JORDAN_A:13

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p, E-max C,C & LE E-max C,q,C holds
Segment p,q,C = (R_Segment (Upper_Arc C),(W-min C),(E-max C),p) \/ (L_Segment (Lower_Arc C),(E-max C),(W-min C),q)

proof end;

theorem Th14: :: JORDAN_A:14

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st LE p, E-max C,C holds
Segment p,(W-min C),C = (R_Segment (Upper_Arc C),(W-min C),(E-max C),p) \/ (L_Segment (Lower_Arc C),(E-max C),(W-min C),(W-min C))

proof end;

theorem Th15: :: JORDAN_A:15

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) holds R_Segment (Upper_Arc C),(W-min C),(E-max C),p = Segment (Upper_Arc C),(W-min C),(E-max C),p,(E-max C)

proof end;

theorem Th16: :: JORDAN_A:16

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) holds L_Segment (Lower_Arc C),(E-max C),(W-min C),p = Segment (Lower_Arc C),(E-max C),(W-min C),(E-max C),p

proof end;

theorem Th17: :: JORDAN_A:17

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st p in C & p <> W-min C holds
Segment p,(W-min C),C is_an_arc_of p, W-min C

proof end;

theorem Th18: :: JORDAN_A:18

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st p <> q & LE p,q,C holds
Segment p,q,C is_an_arc_of p,q

proof end;

theorem Th19: :: JORDAN_A:19

for C being Simple_closed_curve holds C = Segment (W-min C),(W-min C),C

proof end;

theorem Th20: :: JORDAN_A:20

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in C holds
Segment q,(W-min C),C is compact

proof end;

theorem Th21: :: JORDAN_A:21

for C being Simple_closed_curve
for q1, q2 being Point of (TOP-REAL 2) st LE q1,q2,C holds
Segment q1,q2,C is compact

proof end;

definition

let C be Simple_closed_curve;
mode Segmentation of C -> FinSequence of (TOP-REAL 2) means :Def3: :: JORDAN_A:def 3
( it /. 1 = W-min C & it is one-to-one & 8 <= len it & rng it c= C & ( for i being Nat st 1 <= i & i < len it holds
LE it /. i,it /. (i + 1),C ) & ( for i being Nat st 1 <= i & i + 1 < len it holds
(Segment (it /. i),(it /. (i + 1)),C) /\ (Segment (it /. (i + 1)),(it /. (i + 2)),C) = {(it /. (i + 1))} ) & (Segment (it /. (len it)),(it /. 1),C) /\ (Segment (it /. 1),(it /. 2),C) = {(it /. 1)} & (Segment (it /. ((len it) -' 1)),(it /. (len it)),C) /\ (Segment (it /. (len it)),(it /. 1),C) = {(it /. (len it))} & Segment (it /. ((len it) -' 1)),(it /. (len it)),C misses Segment (it /. 1),(it /. 2),C & ( for i, j being Nat st 1 <= i & i < j & j < len it & not i,j are_adjacent1 holds
Segment (it /. i),(it /. (i + 1)),C misses Segment (it /. j),(it /. (j + 1)),C ) & ( for i being Nat st 1 < i & i + 1 < len it holds
Segment (it /. (len it)),(it /. 1),C misses Segment (it /. i),(it /. (i + 1)),C ) );
existence

proof end;

end;

:: deftheorem Def3 defines Segmentation JORDAN_A:def 3 :

registration

let C be Simple_closed_curve;
cluster -> non trivial Segmentation of C;
coherence

proof end;

end;

theorem Th22: :: JORDAN_A:22

for C being Simple_closed_curve
for S being Segmentation of C
for i being Nat st 1 <= i & i <= len S holds
S /. i in C

proof end;

definition

let C be Simple_closed_curve;
let i be natural number ;
let S be Segmentation of C;
func Segm S,i -> Subset of (TOP-REAL 2) equals :Def4: :: JORDAN_A:def 4
Segment (S /. i),(S /. (i + 1)),C if ( 1 <= i & i < len S )
otherwise Segment (S /. (len S)),(S /. 1),C;
correctness ;

end;

:: deftheorem Def4 defines Segm JORDAN_A:def 4 :

theorem :: JORDAN_A:23

for C being Simple_closed_curve
for i being Nat
for S being Segmentation of C st i in dom S holds
Segm S,i c= C

proof end;

registration

let C be Simple_closed_curve;
let S be Segmentation of C;
let i be Nat;
cluster Segm S,i -> non empty compact ;
coherence

proof end;

end;

theorem :: JORDAN_A:24

for C being Simple_closed_curve
for S being Segmentation of C
for p being Point of (TOP-REAL 2) st p in C holds
ex i being natural number st
( i in dom S & p in Segm S,i )

proof end;

theorem Th25: :: JORDAN_A:25

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & not i,j are_adjacent1 holds
Segm S,i misses Segm S,j

proof end;

theorem Th26: :: JORDAN_A:26

for C being Simple_closed_curve
for S being Segmentation of C
for j being Nat st 1 < j & j < (len S) -' 1 holds
Segm S,(len S) misses Segm S,j

proof end;

theorem Th27: :: JORDAN_A:27

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & i,j are_adjacent1 holds
(Segm S,i) /\ (Segm S,j) = {(S /. (i + 1))}

proof end;

theorem Th28: :: JORDAN_A:28

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & i,j are_adjacent1 holds
Segm S,i meets Segm S,j

proof end;

theorem Th29: :: JORDAN_A:29

for C being Simple_closed_curve
for S being Segmentation of C holds (Segm S,(len S)) /\ (Segm S,1) = {(S /. 1)}

proof end;

theorem Th30: :: JORDAN_A:30

for C being Simple_closed_curve
for S being Segmentation of C holds Segm S,(len S) meets Segm S,1

proof end;

theorem Th31: :: JORDAN_A:31

for C being Simple_closed_curve
for S being Segmentation of C holds (Segm S,(len S)) /\ (Segm S,((len S) -' 1)) = {(S /. (len S))}

proof end;

theorem Th32: :: JORDAN_A:32

for C being Simple_closed_curve
for S being Segmentation of C holds Segm S,(len S) meets Segm S,((len S) -' 1)

proof end;

definition

let n be Nat;
let C be Subset of (TOP-REAL n);
func diameter C -> Real means :Def5: :: JORDAN_A:def 5
ex W being Subset of (Euclid n) st
( W = C & it = diameter W );
existence

proof end;

correctness ;

end;

:: deftheorem Def5 defines diameter JORDAN_A:def 5 :

definition

let C be Simple_closed_curve;
let S be Segmentation of C;
func diameter S -> Real means :Def6: :: JORDAN_A:def 6
ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm S,i)) where i is Nat : i in dom S } & it = max S1 );
existence

proof end;

correctness ;

end;

:: deftheorem Def6 defines diameter JORDAN_A:def 6 :

theorem :: JORDAN_A:33

for C being Simple_closed_curve
for S being Segmentation of C
for i being Nat holds diameter (Segm S,i) <= diameter S

proof end;

theorem Th34: :: JORDAN_A:34

for C being Simple_closed_curve
for S being Segmentation of C
for e being Real st ( for i being Nat holds diameter (Segm S,i) < e ) holds
diameter S < e

proof end;

theorem :: JORDAN_A:35

for C being Simple_closed_curve
for e being Real st e > 0 holds
ex S being Segmentation of C st diameter S < e

proof end;

definition

let C be Simple_closed_curve;
let S be Segmentation of C;
func S-Gap S -> Real means :Def7: :: JORDAN_A:def 7
ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min (Segm S,i),(Segm S,j)) where i, j is Nat : ( 1 <= i & i < j & j < len S & not i,j are_adjacent1 ) } & S2 = { (dist_min (Segm S,(len S)),(Segm S,k)) where k is Nat : ( 1 < k & k < (len S) -' 1 ) } & it = min (min S1),(min S2) );
existence

proof end;

correctness ;

end;

:: deftheorem Def7 defines S-Gap JORDAN_A:def 7 :

theorem Th36: :: JORDAN_A:36

for C being Simple_closed_curve
for S being Segmentation of C ex F being non empty finite Subset of REAL st
( F = { (dist_min (Segm S,i),(Segm S,j)) where i, j is Nat : ( 1 <= i & i < j & j <= len S & Segm S,i misses Segm S,j ) } & S-Gap S = min F )

proof end;

theorem :: JORDAN_A:37

for C being Simple_closed_curve
for S being Segmentation of C holds S-Gap S > 0

proof end;