:: TBSP_1 semantic presentation

theorem Th1: :: TBSP_1:1

for L being Real st 0 < L & L < 1 holds
for n, m being Nat st n <= m holds
L to_power m <= L to_power n

proof end;

theorem Th2: :: TBSP_1:2

for L being Real st 0 < L & L < 1 holds
for k being Nat holds
( L to_power k <= 1 & 0 < L to_power k )

proof end;

theorem Th3: :: TBSP_1:3

for L being Real st 0 < L & L < 1 holds
for s being Real st 0 < s holds
ex n being Nat st L to_power n < s

proof end;

definition

let N be non empty MetrStruct ;
attr N is totally_bounded means :Def1: :: TBSP_1:def 1
for r being Real st r > 0 holds
ex G being Subset-Family of N st
( G is finite & the carrier of N = union G & ( for C being Subset of N st C in G holds
ex w being Element of N st C = Ball w,r ) );

end;

:: deftheorem Def1 defines totally_bounded TBSP_1:def 1 :

Lm1: for N being non empty MetrStruct
for f being Function holds
( f is sequence of N iff ( dom f = NAT & ( for x being set st x in NAT holds
f . x is Element of N ) ) )

proof end;

theorem :: TBSP_1:4

canceled;

theorem :: TBSP_1:5

for N being non empty MetrStruct
for f being Function holds
( f is sequence of N iff ( dom f = NAT & ( for n being Nat holds f . n is Element of N ) ) )

proof end;

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
canceled;
attr S2 is convergent means :Def3: :: TBSP_1:def 3
ex x being Element of N st
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st n <= m holds
dist (S2 . m),x < r;

end;

:: deftheorem TBSP_1:def 2 :

:: deftheorem Def3 defines convergent TBSP_1:def 3 :

definition

let M be non empty MetrSpace;
let S1 be sequence of M;
assume A1: S1 is convergent ;
func lim S1 -> Element of M means :: TBSP_1:def 4
for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st m >= n holds
dist (S1 . m),it < r;
existence by A1, Def3;
uniqueness

proof end;

end;

:: deftheorem defines lim TBSP_1:def 4 :

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
attr S2 is Cauchy means :Def5: :: TBSP_1:def 5
for r being Real st r > 0 holds
ex p being Nat st
for n, m being Nat st p <= n & p <= m holds
dist (S2 . n),(S2 . m) < r;

end;

:: deftheorem Def5 defines Cauchy TBSP_1:def 5 :

definition

let N be non empty MetrStruct ;
attr N is complete means :Def6: :: TBSP_1:def 6
for S2 being sequence of N st S2 is Cauchy holds
S2 is convergent;

end;

:: deftheorem Def6 defines complete TBSP_1:def 6 :

theorem :: TBSP_1:6

canceled;

theorem Th7: :: TBSP_1:7

for N being non empty MetrStruct
for S2 being sequence of N st N is triangle & N is symmetric & S2 is convergent holds
S2 is Cauchy

proof end;

registration

let M be non empty symmetric triangle MetrStruct ;
cluster convergent -> Cauchy M5( NAT ,the carrier of M);
coherence by Th7;

end;

theorem Th8: :: TBSP_1:8

for N being non empty MetrStruct
for S2 being sequence of N st N is symmetric holds
( S2 is Cauchy iff for r being Real st r > 0 holds
ex p being Nat st
for n, k being Nat st p <= n holds
dist (S2 . (n + k)),(S2 . n) < r )

proof end;

theorem :: TBSP_1:9

for M being non empty MetrSpace
for f being contraction of M st M is complete holds
ex c being Element of M st
( f . c = c & ( for y being Element of M st f . y = y holds
y = c ) )

proof end;

theorem :: TBSP_1:10

for T being non empty Reflexive symmetric triangle MetrStruct st TopSpaceMetr T is compact holds
T is complete

proof end;

theorem :: TBSP_1:11

canceled;

theorem :: TBSP_1:12

for N being non empty MetrStruct st N is Reflexive & N is triangle & TopSpaceMetr N is compact holds
N is totally_bounded

proof end;

definition

let N be non empty MetrStruct ;
canceled;
attr N is bounded means :Def8: :: TBSP_1:def 8
ex r being Real st
( 0 < r & ( for x, y being Point of N holds dist x,y <= r ) );
let C be Subset of N;
attr C is bounded means :Def9: :: TBSP_1:def 9
ex r being Real st
( 0 < r & ( for x, y being Point of N st x in C & y in C holds
dist x,y <= r ) );

end;

:: deftheorem TBSP_1:def 7 :

:: deftheorem Def8 defines bounded TBSP_1:def 8 :

:: deftheorem Def9 defines bounded TBSP_1:def 9 :

registration

let A be non empty set ;
cluster DiscreteSpace A -> bounded ;
coherence

proof end;

end;

registration

cluster non empty bounded MetrStruct ;
existence

proof end;

end;

theorem :: TBSP_1:13

canceled;

theorem Th14: :: TBSP_1:14

for N being non empty MetrStruct holds {} N is bounded

proof end;

registration

let N be non empty MetrStruct ;
cluster {} N -> bounded ;
coherence by Th14;

end;

registration

let N be non empty MetrStruct ;
cluster bounded Element of K40(the carrier of N);
existence

proof end;

end;

theorem Th15: :: TBSP_1:15

for N being non empty MetrStruct
for C being Subset of N holds
( ( C <> {} & C is bounded implies ex r being Real ex w being Element of N st
( 0 < r & w in C & ( for z being Point of N st z in C holds
dist w,z <= r ) ) ) & ( N is symmetric & N is triangle & ex r being Real ex w being Element of N st
( 0 < r & w in C & ( for z being Point of N st z in C holds
dist w,z <= r ) ) implies C is bounded ) )

proof end;

theorem Th16: :: TBSP_1:16

for N being non empty MetrStruct
for w being Element of N
for r being real number st N is Reflexive & 0 < r holds
w in Ball w,r

proof end;

theorem Th17: :: TBSP_1:17

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being real number st r <= 0 holds
Ball t1,r = {}

proof end;

Lm2: for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being real number st 0 < r holds
Ball t1,r is bounded

proof end;

theorem :: TBSP_1:18

canceled;

theorem Th19: :: TBSP_1:19

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being real number holds Ball t1,r is bounded

proof end;

theorem Th20: :: TBSP_1:20

for T being non empty Reflexive symmetric triangle MetrStruct
for P, Q being Subset of T st P is bounded & Q is bounded holds
P \/ Q is bounded

proof end;

theorem Th21: :: TBSP_1:21

for N being non empty MetrStruct
for C, D being Subset of N st C is bounded & D c= C holds
D is bounded

proof end;

theorem Th22: :: TBSP_1:22

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for P being Subset of T st P = {t1} holds
P is bounded

proof end;

theorem Th23: :: TBSP_1:23

for T being non empty Reflexive symmetric triangle MetrStruct
for P being Subset of T st P is finite holds
P is bounded

proof end;

registration

let T be non empty Reflexive symmetric triangle MetrStruct ;
cluster finite -> bounded Element of K40(the carrier of T);
coherence by Th23;

end;

registration

let T be non empty Reflexive symmetric triangle MetrStruct ;
cluster non empty finite bounded Element of K40(the carrier of T);
existence

proof end;

end;

theorem Th24: :: TBSP_1:24

for T being non empty Reflexive symmetric triangle MetrStruct
for Y being Subset-Family of T st Y is finite & ( for P being Subset of T st P in Y holds
P is bounded ) holds
union Y is bounded

proof end;

theorem Th25: :: TBSP_1:25

for N being non empty MetrStruct holds
( N is bounded iff [#] N is bounded )

proof end;

registration

let N be non empty bounded MetrStruct ;
cluster [#] N -> bounded ;
coherence by Th25;

end;

theorem :: TBSP_1:26

for T being non empty Reflexive symmetric triangle MetrStruct st T is totally_bounded holds
T is bounded

proof end;

definition

let N be non empty Reflexive MetrStruct ;
let C be Subset of N;
assume A1: C is bounded ;
func diameter C -> Real means :Def10: :: TBSP_1:def 10
( ( for x, y being Point of N st x in C & y in C holds
dist x,y <= it ) & ( for s being Real st ( for x, y being Point of N st x in C & y in C holds
dist x,y <= s ) holds
it <= s ) ) if C <> {}
otherwise it = 0;
consistency ;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def10 defines diameter TBSP_1:def 10 :

theorem :: TBSP_1:27

canceled;

theorem :: TBSP_1:28

for T being non empty Reflexive symmetric triangle MetrStruct
for x being set
for P being Subset of T st P = {x} holds
diameter P = 0

proof end;

theorem Th29: :: TBSP_1:29

for R being non empty Reflexive MetrStruct
for S being Subset of R st S is bounded holds
0 <= diameter S

proof end;

theorem :: TBSP_1:30

for M being non empty MetrSpace
for A being Subset of M st A <> {} & A is bounded & diameter A = 0 holds
ex g being Point of M st A = {g}

proof end;

theorem :: TBSP_1:31

for T being non empty Reflexive symmetric triangle MetrStruct
for t1 being Element of T
for r being Real st 0 < r holds
diameter (Ball t1,r) <= 2 * r

proof end;

theorem :: TBSP_1:32

for R being non empty Reflexive MetrStruct
for S, V being Subset of R st S is bounded & V c= S holds
diameter V <= diameter S

proof end;

theorem :: TBSP_1:33

for T being non empty Reflexive symmetric triangle MetrStruct
for P, Q being Subset of T st P is bounded & Q is bounded & P meets Q holds
diameter (P \/ Q) <= (diameter P) + (diameter Q)

proof end;

definition

let N be non empty MetrStruct ;
let S2 be sequence of N;
:: original: rng
redefine func rng S2 -> Subset of N;
coherence by RELSET_1:12;

end;

theorem :: TBSP_1:34

for T being non empty Reflexive symmetric triangle MetrStruct
for S1 being sequence of T st S1 is Cauchy holds
rng S1 is bounded

proof end;