:: by Jan Popio{\l}ek

::

:: Received July 19, 1991

:: Copyright (c) 1991-2012 Association of Mizar Users

begin

deffunc H

:: deftheorem Def1 defines convergent BHSP_2:def 1 :

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is convergent iff ex g being Point of X st

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),g) < r );

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is convergent iff ex g being Point of X st

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),g) < r );

theorem Th2: :: BHSP_2:2

for X being RealUnitarySpace

for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st

for n being Element of NAT st k <= n holds

seq9 . n = seq . n holds

seq9 is convergent

for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st

for n being Element of NAT st k <= n holds

seq9 . n = seq . n holds

seq9 is convergent

proof end;

theorem Th3: :: BHSP_2:3

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

seq1 + seq2 is convergent

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

seq1 + seq2 is convergent

proof end;

theorem Th4: :: BHSP_2:4

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

seq1 - seq2 is convergent

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

seq1 - seq2 is convergent

proof end;

theorem Th5: :: BHSP_2:5

for X being RealUnitarySpace

for a being Real

for seq being sequence of X st seq is convergent holds

a * seq is convergent

for a being Real

for seq being sequence of X st seq is convergent holds

a * seq is convergent

proof end;

theorem Th6: :: BHSP_2:6

for X being RealUnitarySpace

for seq being sequence of X st seq is convergent holds

- seq is convergent

for seq being sequence of X st seq is convergent holds

- seq is convergent

proof end;

theorem Th7: :: BHSP_2:7

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is convergent holds

seq + x is convergent

for x being Point of X

for seq being sequence of X st seq is convergent holds

seq + x is convergent

proof end;

theorem Th8: :: BHSP_2:8

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is convergent holds

seq - x is convergent

for x being Point of X

for seq being sequence of X st seq is convergent holds

seq - x is convergent

proof end;

theorem Th9: :: BHSP_2:9

for X being RealUnitarySpace

for seq being sequence of X holds

( seq is convergent iff ex g being Point of X st

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq . n) - g).|| < r )

for seq being sequence of X holds

( seq is convergent iff ex g being Point of X st

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq . n) - g).|| < r )

proof end;

definition

let X be RealUnitarySpace;

let seq be sequence of X;

assume A1: seq is convergent ;

ex b_{1} being Point of X st

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b_{1}) < r
by A1, Def1;

uniqueness

for b_{1}, b_{2} being Point of X st ( for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b_{1}) < r ) & ( for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b_{2}) < r ) holds

b_{1} = b_{2}

end;
let seq be sequence of X;

assume A1: seq is convergent ;

func lim seq -> Point of X means :Def2: :: BHSP_2:def 2

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),it) < r;

existence for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),it) < r;

ex b

for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b

uniqueness

for b

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b

b

proof end;

:: deftheorem Def2 defines lim BHSP_2:def 2 :

for X being RealUnitarySpace

for seq being sequence of X st seq is convergent holds

for b_{3} being Point of X holds

( b_{3} = lim seq iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b_{3}) < r );

for X being RealUnitarySpace

for seq being sequence of X st seq is convergent holds

for b

( b

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

dist ((seq . n),b

theorem Th10: :: BHSP_2:10

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is constant & x in rng seq holds

lim seq = x

for x being Point of X

for seq being sequence of X st seq is constant & x in rng seq holds

lim seq = x

proof end;

theorem :: BHSP_2:11

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is constant & ex n being Element of NAT st seq . n = x holds

lim seq = x

for x being Point of X

for seq being sequence of X st seq is constant & ex n being Element of NAT st seq . n = x holds

lim seq = x

proof end;

theorem :: BHSP_2:12

for X being RealUnitarySpace

for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st

for n being Element of NAT st n >= k holds

seq9 . n = seq . n holds

lim seq = lim seq9

for seq, seq9 being sequence of X st seq is convergent & ex k being Element of NAT st

for n being Element of NAT st n >= k holds

seq9 . n = seq . n holds

lim seq = lim seq9

proof end;

theorem Th13: :: BHSP_2:13

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

lim (seq1 + seq2) = (lim seq1) + (lim seq2)

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

lim (seq1 + seq2) = (lim seq1) + (lim seq2)

proof end;

theorem Th14: :: BHSP_2:14

for X being RealUnitarySpace

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

lim (seq1 - seq2) = (lim seq1) - (lim seq2)

for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds

lim (seq1 - seq2) = (lim seq1) - (lim seq2)

proof end;

theorem Th15: :: BHSP_2:15

for X being RealUnitarySpace

for a being Real

for seq being sequence of X st seq is convergent holds

lim (a * seq) = a * (lim seq)

for a being Real

for seq being sequence of X st seq is convergent holds

lim (a * seq) = a * (lim seq)

proof end;

theorem Th16: :: BHSP_2:16

for X being RealUnitarySpace

for seq being sequence of X st seq is convergent holds

lim (- seq) = - (lim seq)

for seq being sequence of X st seq is convergent holds

lim (- seq) = - (lim seq)

proof end;

theorem Th17: :: BHSP_2:17

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is convergent holds

lim (seq + x) = (lim seq) + x

for x being Point of X

for seq being sequence of X st seq is convergent holds

lim (seq + x) = (lim seq) + x

proof end;

theorem :: BHSP_2:18

for X being RealUnitarySpace

for x being Point of X

for seq being sequence of X st seq is convergent holds

lim (seq - x) = (lim seq) - x

for x being Point of X

for seq being sequence of X st seq is convergent holds

lim (seq - x) = (lim seq) - x

proof end;

theorem Th19: :: BHSP_2:19

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent holds

( lim seq = g iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq . n) - g).|| < r )

for g being Point of X

for seq being sequence of X st seq is convergent holds

( lim seq = g iff for r being Real st r > 0 holds

ex m being Element of NAT st

for n being Element of NAT st n >= m holds

||.((seq . n) - g).|| < r )

proof end;

definition

let X be RealUnitarySpace;

let seq be sequence of X;

ex b_{1} being Real_Sequence st

for n being Element of NAT holds b_{1} . n = ||.(seq . n).||

for b_{1}, b_{2} being Real_Sequence st ( for n being Element of NAT holds b_{1} . n = ||.(seq . n).|| ) & ( for n being Element of NAT holds b_{2} . n = ||.(seq . n).|| ) holds

b_{1} = b_{2}

end;
let seq be sequence of X;

func ||.seq.|| -> Real_Sequence means :Def3: :: BHSP_2:def 3

for n being Element of NAT holds it . n = ||.(seq . n).||;

existence for n being Element of NAT holds it . n = ||.(seq . n).||;

ex b

for n being Element of NAT holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def3 defines ||. BHSP_2:def 3 :

for X being RealUnitarySpace

for seq being sequence of X

for b_{3} being Real_Sequence holds

( b_{3} = ||.seq.|| iff for n being Element of NAT holds b_{3} . n = ||.(seq . n).|| );

for X being RealUnitarySpace

for seq being sequence of X

for b

( b

theorem Th20: :: BHSP_2:20

for X being RealUnitarySpace

for seq being sequence of X st seq is convergent holds

||.seq.|| is convergent

for seq being sequence of X st seq is convergent holds

||.seq.|| is convergent

proof end;

theorem Th21: :: BHSP_2:21

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.seq.|| is convergent & lim ||.seq.|| = ||.g.|| )

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.seq.|| is convergent & lim ||.seq.|| = ||.g.|| )

proof end;

theorem Th22: :: BHSP_2:22

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 )

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(seq - g).|| is convergent & lim ||.(seq - g).|| = 0 )

proof end;

definition

let X be RealUnitarySpace;

let seq be sequence of X;

let x be Point of X;

ex b_{1} being Real_Sequence st

for n being Element of NAT holds b_{1} . n = dist ((seq . n),x)

for b_{1}, b_{2} being Real_Sequence st ( for n being Element of NAT holds b_{1} . n = dist ((seq . n),x) ) & ( for n being Element of NAT holds b_{2} . n = dist ((seq . n),x) ) holds

b_{1} = b_{2}

end;
let seq be sequence of X;

let x be Point of X;

func dist (seq,x) -> Real_Sequence means :Def4: :: BHSP_2:def 4

for n being Element of NAT holds it . n = dist ((seq . n),x);

existence for n being Element of NAT holds it . n = dist ((seq . n),x);

ex b

for n being Element of NAT holds b

proof end;

uniqueness for b

b

proof end;

:: deftheorem Def4 defines dist BHSP_2:def 4 :

for X being RealUnitarySpace

for seq being sequence of X

for x being Point of X

for b_{4} being Real_Sequence holds

( b_{4} = dist (seq,x) iff for n being Element of NAT holds b_{4} . n = dist ((seq . n),x) );

for X being RealUnitarySpace

for seq being sequence of X

for x being Point of X

for b

( b

theorem Th23: :: BHSP_2:23

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

dist (seq,g) is convergent

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

dist (seq,g) is convergent

proof end;

theorem Th24: :: BHSP_2:24

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist (seq,g) is convergent & lim (dist (seq,g)) = 0 )

proof end;

theorem :: BHSP_2:25

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.(seq1 + seq2).|| is convergent & lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.(seq1 + seq2).|| is convergent & lim ||.(seq1 + seq2).|| = ||.(g1 + g2).|| )

proof end;

theorem :: BHSP_2:26

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.((seq1 + seq2) - (g1 + g2)).|| is convergent & lim ||.((seq1 + seq2) - (g1 + g2)).|| = 0 )

proof end;

theorem :: BHSP_2:27

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.(seq1 - seq2).|| is convergent & lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.(seq1 - seq2).|| is convergent & lim ||.(seq1 - seq2).|| = ||.(g1 - g2).|| )

proof end;

theorem :: BHSP_2:28

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent & lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( ||.((seq1 - seq2) - (g1 - g2)).|| is convergent & lim ||.((seq1 - seq2) - (g1 - g2)).|| = 0 )

proof end;

theorem :: BHSP_2:29

for X being RealUnitarySpace

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(a * seq).|| is convergent & lim ||.(a * seq).|| = ||.(a * g).|| )

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(a * seq).|| is convergent & lim ||.(a * seq).|| = ||.(a * g).|| )

proof end;

theorem :: BHSP_2:30

for X being RealUnitarySpace

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((a * seq) - (a * g)).|| is convergent & lim ||.((a * seq) - (a * g)).|| = 0 )

proof end;

theorem :: BHSP_2:31

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(- seq).|| is convergent & lim ||.(- seq).|| = ||.(- g).|| )

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(- seq).|| is convergent & lim ||.(- seq).|| = ||.(- g).|| )

proof end;

theorem :: BHSP_2:32

for X being RealUnitarySpace

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((- seq) - (- g)).|| is convergent & lim ||.((- seq) - (- g)).|| = 0 )

for g being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((- seq) - (- g)).|| is convergent & lim ||.((- seq) - (- g)).|| = 0 )

proof end;

Lm1: for X being RealUnitarySpace

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(seq + x).|| is convergent & lim ||.(seq + x).|| = ||.(g + x).|| )

proof end;

theorem Th33: :: BHSP_2:33

for X being RealUnitarySpace

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((seq + x) - (g + x)).|| is convergent & lim ||.((seq + x) - (g + x)).|| = 0 )

proof end;

theorem :: BHSP_2:34

for X being RealUnitarySpace

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(seq - x).|| is convergent & lim ||.(seq - x).|| = ||.(g - x).|| )

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.(seq - x).|| is convergent & lim ||.(seq - x).|| = ||.(g - x).|| )

proof end;

theorem :: BHSP_2:35

for X being RealUnitarySpace

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((seq - x) - (g - x)).|| is convergent & lim ||.((seq - x) - (g - x)).|| = 0 )

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( ||.((seq - x) - (g - x)).|| is convergent & lim ||.((seq - x) - (g - x)).|| = 0 )

proof end;

theorem :: BHSP_2:36

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( dist ((seq1 + seq2),(g1 + g2)) is convergent & lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( dist ((seq1 + seq2),(g1 + g2)) is convergent & lim (dist ((seq1 + seq2),(g1 + g2))) = 0 )

proof end;

theorem :: BHSP_2:37

for X being RealUnitarySpace

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( dist ((seq1 - seq2),(g1 - g2)) is convergent & lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )

for g1, g2 being Point of X

for seq1, seq2 being sequence of X st seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2 holds

( dist ((seq1 - seq2),(g1 - g2)) is convergent & lim (dist ((seq1 - seq2),(g1 - g2))) = 0 )

proof end;

theorem :: BHSP_2:38

for X being RealUnitarySpace

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist ((a * seq),(a * g)) is convergent & lim (dist ((a * seq),(a * g))) = 0 )

for g being Point of X

for a being Real

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist ((a * seq),(a * g)) is convergent & lim (dist ((a * seq),(a * g))) = 0 )

proof end;

theorem :: BHSP_2:39

for X being RealUnitarySpace

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist ((seq + x),(g + x)) is convergent & lim (dist ((seq + x),(g + x))) = 0 )

for g, x being Point of X

for seq being sequence of X st seq is convergent & lim seq = g holds

( dist ((seq + x),(g + x)) is convergent & lim (dist ((seq + x),(g + x))) = 0 )

proof end;

definition

let X be RealUnitarySpace;

let x be Point of X;

let r be Real;

{ y where y is Point of X : ||.(x - y).|| < r } is Subset of X

{ y where y is Point of X : ||.(x - y).|| <= r } is Subset of X

{ y where y is Point of X : ||.(x - y).|| = r } is Subset of X

end;
let x be Point of X;

let r be Real;

func Ball (x,r) -> Subset of X equals :: BHSP_2:def 5

{ y where y is Point of X : ||.(x - y).|| < r } ;

coherence { y where y is Point of X : ||.(x - y).|| < r } ;

{ y where y is Point of X : ||.(x - y).|| < r } is Subset of X

proof end;

func cl_Ball (x,r) -> Subset of X equals :: BHSP_2:def 6

{ y where y is Point of X : ||.(x - y).|| <= r } ;

coherence { y where y is Point of X : ||.(x - y).|| <= r } ;

{ y where y is Point of X : ||.(x - y).|| <= r } is Subset of X

proof end;

func Sphere (x,r) -> Subset of X equals :: BHSP_2:def 7

{ y where y is Point of X : ||.(x - y).|| = r } ;

coherence { y where y is Point of X : ||.(x - y).|| = r } ;

{ y where y is Point of X : ||.(x - y).|| = r } is Subset of X

proof end;

:: deftheorem defines Ball BHSP_2:def 5 :

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Ball (x,r) = { y where y is Point of X : ||.(x - y).|| < r } ;

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Ball (x,r) = { y where y is Point of X : ||.(x - y).|| < r } ;

:: deftheorem defines cl_Ball BHSP_2:def 6 :

for X being RealUnitarySpace

for x being Point of X

for r being Real holds cl_Ball (x,r) = { y where y is Point of X : ||.(x - y).|| <= r } ;

for X being RealUnitarySpace

for x being Point of X

for r being Real holds cl_Ball (x,r) = { y where y is Point of X : ||.(x - y).|| <= r } ;

:: deftheorem defines Sphere BHSP_2:def 7 :

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Sphere (x,r) = { y where y is Point of X : ||.(x - y).|| = r } ;

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Sphere (x,r) = { y where y is Point of X : ||.(x - y).|| = r } ;

theorem Th40: :: BHSP_2:40

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

for z, x being Point of X

for r being Real holds

( z in Ball (x,r) iff ||.(x - z).|| < r )

proof end;

theorem Th41: :: BHSP_2:41

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in Ball (x,r) iff dist (x,z) < r )

for z, x being Point of X

for r being Real holds

( z in Ball (x,r) iff dist (x,z) < r )

proof end;

theorem :: BHSP_2:43

for X being RealUnitarySpace

for y, x, z being Point of X

for r being Real st y in Ball (x,r) & z in Ball (x,r) holds

dist (y,z) < 2 * r

for y, x, z being Point of X

for r being Real st y in Ball (x,r) & z in Ball (x,r) holds

dist (y,z) < 2 * r

proof end;

theorem :: BHSP_2:44

for X being RealUnitarySpace

for y, x, z being Point of X

for r being Real st y in Ball (x,r) holds

y - z in Ball ((x - z),r)

for y, x, z being Point of X

for r being Real st y in Ball (x,r) holds

y - z in Ball ((x - z),r)

proof end;

theorem :: BHSP_2:45

for X being RealUnitarySpace

for y, x being Point of X

for r being Real st y in Ball (x,r) holds

y - x in Ball ((0. X),r)

for y, x being Point of X

for r being Real st y in Ball (x,r) holds

y - x in Ball ((0. X),r)

proof end;

theorem :: BHSP_2:46

for X being RealUnitarySpace

for y, x being Point of X

for r, q being Real st y in Ball (x,r) & r <= q holds

y in Ball (x,q)

for y, x being Point of X

for r, q being Real st y in Ball (x,r) & r <= q holds

y in Ball (x,q)

proof end;

theorem Th47: :: BHSP_2:47

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in cl_Ball (x,r) iff ||.(x - z).|| <= r )

for z, x being Point of X

for r being Real holds

( z in cl_Ball (x,r) iff ||.(x - z).|| <= r )

proof end;

theorem Th48: :: BHSP_2:48

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in cl_Ball (x,r) iff dist (x,z) <= r )

for z, x being Point of X

for r being Real holds

( z in cl_Ball (x,r) iff dist (x,z) <= r )

proof end;

theorem :: BHSP_2:49

for X being RealUnitarySpace

for x being Point of X

for r being Real st r >= 0 holds

x in cl_Ball (x,r)

for x being Point of X

for r being Real st r >= 0 holds

x in cl_Ball (x,r)

proof end;

theorem Th50: :: BHSP_2:50

for X being RealUnitarySpace

for y, x being Point of X

for r being Real st y in Ball (x,r) holds

y in cl_Ball (x,r)

for y, x being Point of X

for r being Real st y in Ball (x,r) holds

y in cl_Ball (x,r)

proof end;

theorem Th51: :: BHSP_2:51

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in Sphere (x,r) iff ||.(x - z).|| = r )

for z, x being Point of X

for r being Real holds

( z in Sphere (x,r) iff ||.(x - z).|| = r )

proof end;

theorem :: BHSP_2:52

for X being RealUnitarySpace

for z, x being Point of X

for r being Real holds

( z in Sphere (x,r) iff dist (x,z) = r )

for z, x being Point of X

for r being Real holds

( z in Sphere (x,r) iff dist (x,z) = r )

proof end;

theorem :: BHSP_2:53

for X being RealUnitarySpace

for y, x being Point of X

for r being Real st y in Sphere (x,r) holds

y in cl_Ball (x,r)

for y, x being Point of X

for r being Real st y in Sphere (x,r) holds

y in cl_Ball (x,r)

proof end;

theorem Th54: :: BHSP_2:54

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Ball (x,r) c= cl_Ball (x,r)

for x being Point of X

for r being Real holds Ball (x,r) c= cl_Ball (x,r)

proof end;

theorem Th55: :: BHSP_2:55

for X being RealUnitarySpace

for x being Point of X

for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

for x being Point of X

for r being Real holds Sphere (x,r) c= cl_Ball (x,r)

proof end;

theorem :: BHSP_2:56

for X being RealUnitarySpace

for x being Point of X

for r being Real holds (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r)

for x being Point of X

for r being Real holds (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r)

proof end;