:: Some Properties of Dyadic Numbers and Intervals
:: by J\'ozef Bia{\l}as and Yatsuka Nakamura
::
:: Received February 16, 2001
:: Copyright (c) 2001-2011 Association of Mizar Users

begin

theorem :: URYSOHN2:1

canceled;

theorem Th2: :: URYSOHN2:2

for A being Subset of REAL
for x being Real st x <> 0 holds
(x ") ** (x ** A) = A

proof end;

theorem Th3: :: URYSOHN2:3

for x being Real st x <> 0 holds
for A being Subset of REAL st A = REAL holds
x ** A = A

proof end;

theorem Th4: :: URYSOHN2:4

for A being Subset of REAL st A <> {} holds
0 ** A = {0}

proof end;

theorem :: URYSOHN2:5

for x being Real holds x ** {} = {} ;

theorem Th6: :: URYSOHN2:6

for a, b being R_eal holds
( not a <= b or ( a = -infty & b = -infty ) or ( a = -infty & b in REAL ) or ( a = -infty & b = +infty ) or ( a in REAL & b in REAL ) or ( a in REAL & b = +infty ) or ( a = +infty & b = +infty ) )

proof end;

theorem :: URYSOHN2:7

canceled;

theorem Th8: :: URYSOHN2:8

for A being Interval holds 0 ** A is interval

proof end;

theorem Th9: :: URYSOHN2:9

for A being non empty Interval
for x being Real st x <> 0 & A is open_interval holds
x ** A is open_interval

proof end;

theorem Th10: :: URYSOHN2:10

for A being non empty Interval
for x being Real st x <> 0 & A is closed_interval holds
x ** A is closed_interval

proof end;

theorem Th11: :: URYSOHN2:11

for A being non empty Interval
for x being Real st 0 < x & A is right_open_interval holds
x ** A is right_open_interval

proof end;

theorem Th12: :: URYSOHN2:12

for A being non empty Interval
for x being Real st x < 0 & A is right_open_interval holds
x ** A is left_open_interval

proof end;

theorem Th13: :: URYSOHN2:13

for A being non empty Interval
for x being Real st 0 < x & A is left_open_interval holds
x ** A is left_open_interval

proof end;

theorem Th14: :: URYSOHN2:14

for A being non empty Interval
for x being Real st x < 0 & A is left_open_interval holds
x ** A is right_open_interval

proof end;

theorem :: URYSOHN2:15

for A being non empty Interval
for x being Real st 0 < x holds
for B being non empty Interval st B = x ** A & A = [.(inf A),(sup A).] holds
( B = [.(inf B),(sup B).] & ( for s, t being Real st s = inf A & t = sup A holds
( inf B = x * s & sup B = x * t ) ) )

proof end;

theorem :: URYSOHN2:16

for A being non empty Interval
for x being Real st 0 < x holds
for B being non empty Interval st B = x ** A & A = ].(inf A),(sup A).] holds
( B = ].(inf B),(sup B).] & ( for s, t being Real st s = inf A & t = sup A holds
( inf B = x * s & sup B = x * t ) ) )

proof end;

theorem :: URYSOHN2:17

for A being non empty Interval
for x being Real st 0 < x holds
for B being non empty Interval st B = x ** A & A = ].(inf A),(sup A).[ holds
( B = ].(inf B),(sup B).[ & ( for s, t being Real st s = inf A & t = sup A holds
( inf B = x * s & sup B = x * t ) ) )

proof end;

theorem :: URYSOHN2:18

for A being non empty Interval
for x being Real st 0 < x holds
for B being non empty Interval st B = x ** A & A = [.(inf A),(sup A).[ holds
( B = [.(inf B),(sup B).[ & ( for s, t being Real st s = inf A & t = sup A holds
( inf B = x * s & sup B = x * t ) ) )

proof end;

theorem Th19: :: URYSOHN2:19

for A being non empty Interval
for x being Real holds x ** A is Interval

proof end;

registration

let A be interval Subset of REAL;
let x be Real;
cluster x ** A -> interval ;
correctness
coherence
x ** A is interval ;

proof end;

end;

Lm1: for A being non empty Subset of REAL
for x being Real st x > 0 & x ** A is bounded_above holds
A is bounded_above

proof end;

theorem Th20: :: URYSOHN2:20

for A being non empty Subset of REAL
for x being Real
for y being R_eal st x = y & 0 <= y holds
sup (x ** A) = y * (sup A)

proof end;

Lm2: for A being non empty Subset of REAL
for x being Real st x > 0 & x ** A is bounded_below holds
A is bounded_below

proof end;

theorem Th21: :: URYSOHN2:21

for A being non empty Subset of REAL
for x being Real
for y being R_eal st x = y & 0 <= y holds
inf (x ** A) = y * (inf A)

proof end;

theorem :: URYSOHN2:22

for A being Interval
for x being Real st 0 <= x holds
for y being Real st y = diameter A holds
x * y = diameter (x ** A)

proof end;

theorem Th23: :: URYSOHN2:23

for eps being Real st 0 < eps holds
ex n being Element of NAT st 1 < (2 |^ n) * eps

proof end;

theorem Th24: :: URYSOHN2:24

for a, b being Real st 0 <= a & 1 < b - a holds
ex n being Element of NAT st
( a < n & n < b )

proof end;

theorem :: URYSOHN2:25

canceled;

theorem :: URYSOHN2:26

canceled;

theorem :: URYSOHN2:27

for n being Element of NAT holds dyadic n c= DYADIC

proof end;

theorem Th28: :: URYSOHN2:28

for a, b being Real st a < b & 0 <= a & b <= 1 holds
ex c being Real st
( c in DYADIC & a < c & c < b )

proof end;

theorem Th29: :: URYSOHN2:29

for a, b being Real st a < b holds
ex c being Real st
( c in DOM & a < c & c < b )

proof end;

theorem :: URYSOHN2:30

for A being non empty Subset of ExtREAL
for a, b being R_eal st A c= [.a,b.] holds
( a <= inf A & sup A <= b )

proof end;

theorem :: URYSOHN2:31

( 0 in DYADIC & 1 in DYADIC )

proof end;

theorem :: URYSOHN2:32

DYADIC c= [.0,1.]

proof end;

theorem :: URYSOHN2:33

for n, k being Element of NAT st n <= k holds
dyadic n c= dyadic k

proof end;

theorem Th34: :: URYSOHN2:34

for a, b, c, d being Real st a < c & c < b & a < d & d < b holds
abs (d - c) < b - a

proof end;

theorem :: URYSOHN2:35

for eps being Real st 0 < eps holds
for d being Real st 0 < d holds
ex r1, r2 being Real st
( r1 in DYADIC \/ (right_open_halfline 1) & r2 in DYADIC \/ (right_open_halfline 1) & 0 < r1 & r1 < d & d < r2 & r2 - r1 < eps )

proof end;

begin

theorem :: URYSOHN2:36

for A being non empty Subset of REAL
for x being Real st x > 0 & x ** A is bounded_above holds
A is bounded_above by Lm1;

theorem :: URYSOHN2:37

for A being non empty Subset of REAL
for x being Real st x > 0 & x ** A is bounded_below holds
A is bounded_below by Lm2;