:: Bounding boxes for compact sets in ${\calE}^2$
:: by Czes{\l}aw Byli\'nski and Piotr Rudnicki
::
:: Received July 29, 1997
:: Copyright (c) 1997-2019 Association of Mizar Users

Lm1: for X being Subset of REAL st X is bounded_below holds
-- X is bounded_above

proof end;

Lm2: for X being non empty set
for f being Function of X,REAL st f is with_min holds
- f is with_max

proof end;

definition
let T be 1-sorted ;
mode RealMap of T is Function of the carrier of T,REAL;
end;

registration
let T be non empty 1-sorted ;
cluster non empty Relation-like the carrier of T -defined REAL -valued Function-like V31( the carrier of T) V32( the carrier of T, REAL ) complex-valued ext-real-valued real-valued bounded for Element of K32(K33( the carrier of T,REAL));
existence
ex b1 being RealMap of T st b1 is bounded
proof end;
end;

definition
let T be 1-sorted ;
let f be RealMap of T;
func lower_bound f -> Real equals :: PSCOMP_1:def 1
lower_bound (f .: the carrier of T);
coherence
lower_bound (f .: the carrier of T) is Real
;
func upper_bound f -> Real equals :: PSCOMP_1:def 2
upper_bound (f .: the carrier of T);
coherence
upper_bound (f .: the carrier of T) is Real
;
end;

:: deftheorem defines lower_bound PSCOMP_1:def 1 :
for T being 1-sorted
for f being RealMap of T holds lower_bound f = lower_bound (f .: the carrier of T);

:: deftheorem defines upper_bound PSCOMP_1:def 2 :
for T being 1-sorted
for f being RealMap of T holds upper_bound f = upper_bound (f .: the carrier of T);

theorem Th1: :: PSCOMP_1:1
for T being non empty TopSpace
for f being V50() RealMap of T
for p being Point of T holds f . p >= lower_bound f
proof end;

theorem :: PSCOMP_1:2
for T being non empty TopSpace
for f being V50() RealMap of T
for s being Real st ( for t being Point of T holds f . t >= s ) holds
lower_bound f >= s
proof end;

theorem :: PSCOMP_1:3
for r being Real
for T being non empty TopSpace
for f being RealMap of T st ( for p being Point of T holds f . p >= r ) & ( for t being Real st ( for p being Point of T holds f . p >= t ) holds
r >= t ) holds
r = lower_bound f
proof end;

theorem Th4: :: PSCOMP_1:4
for T being non empty TopSpace
for f being V49() RealMap of T
for p being Point of T holds f . p <= upper_bound f
proof end;

theorem :: PSCOMP_1:5
for T being non empty TopSpace
for f being V49() RealMap of T
for t being Real st ( for p being Point of T holds f . p <= t ) holds
upper_bound f <= t
proof end;

theorem :: PSCOMP_1:6
for r being Real
for T being non empty TopSpace
for f being RealMap of T st ( for p being Point of T holds f . p <= r ) & ( for t being Real st ( for p being Point of T holds f . p <= t ) holds
r <= t ) holds
r = upper_bound f
proof end;

theorem Th7: :: PSCOMP_1:7
for T being non empty 1-sorted
for f being bounded RealMap of T holds lower_bound f <= upper_bound f
proof end;

definition
let T be TopStruct ;
let f be RealMap of T;
attr f is continuous means :: PSCOMP_1:def 3
for Y being Subset of REAL st Y is closed holds
f " Y is closed ;
end;

:: deftheorem defines continuous PSCOMP_1:def 3 :
for T being TopStruct
for f being RealMap of T holds
( f is continuous iff for Y being Subset of REAL st Y is closed holds
f " Y is closed );

registration
let T be non empty TopSpace;
cluster non empty Relation-like the carrier of T -defined REAL -valued Function-like V31( the carrier of T) V32( the carrier of T, REAL ) complex-valued ext-real-valued real-valued continuous for Element of K32(K33( the carrier of T,REAL));
existence
ex b1 being RealMap of T st b1 is continuous
proof end;
end;

registration
let T be non empty TopSpace;
let S be non empty SubSpace of T;
cluster non empty Relation-like the carrier of S -defined REAL -valued Function-like V31( the carrier of S) V32( the carrier of S, REAL ) complex-valued ext-real-valued real-valued continuous for Element of K32(K33( the carrier of S,REAL));
existence
ex b1 being RealMap of S st b1 is continuous
proof end;
end;

theorem Th8: :: PSCOMP_1:8
for T being TopStruct
for f being RealMap of T holds
( f is continuous iff for Y being Subset of REAL st Y is open holds
f " Y is open )
proof end;

theorem Th9: :: PSCOMP_1:9
for T being TopStruct
for f being RealMap of T st f is continuous holds
- f is continuous
proof end;

theorem Th10: :: PSCOMP_1:10
for r3 being Real
for T being TopStruct
for f being RealMap of T st f is continuous holds
r3 + f is continuous
proof end;

theorem Th11: :: PSCOMP_1:11
for T being TopStruct
for f being RealMap of T st f is continuous & not 0 in rng f holds
Inv f is continuous
proof end;

theorem :: PSCOMP_1:12
for T being TopStruct
for f being RealMap of T
for R being Subset-Family of REAL st f is continuous & R is open holds
(" f) .: R is open
proof end;

theorem Th13: :: PSCOMP_1:13
for T being TopStruct
for f being RealMap of T
for R being Subset-Family of REAL st f is continuous & R is closed holds
(" f) .: R is closed
proof end;

definition
let T be non empty TopStruct ;
let f be RealMap of T;
let X be Subset of T;
:: original: |
redefine func f | X -> RealMap of (T | X);
coherence
f | X is RealMap of (T | X)
proof end;
end;

registration
let T be non empty TopSpace;
let f be continuous RealMap of T;
let X be Subset of T;
cluster f | X -> continuous for RealMap of (T | X);
coherence
for b1 being RealMap of (T | X) st b1 = f | X holds
b1 is continuous
proof end;
end;

registration
let T be non empty TopSpace;
let P be non empty compact Subset of T;
cluster T | P -> compact ;
coherence
T | P is compact
by COMPTS_1:3;
end;

theorem Th14: :: PSCOMP_1:14
for T being non empty TopSpace holds
( ( for f being RealMap of T st f is continuous holds
f is with_max ) iff for f being RealMap of T st f is continuous holds
f is with_min )
proof end;

theorem Th15: :: PSCOMP_1:15
for T being non empty TopSpace holds
( ( for f being RealMap of T st f is continuous holds
f is bounded ) iff for f being RealMap of T st f is continuous holds
f is with_max )
proof end;

definition
let T be TopStruct ;
attr T is pseudocompact means :Def4: :: PSCOMP_1:def 4
for f being RealMap of T st f is continuous holds
f is bounded ;
end;

:: deftheorem Def4 defines pseudocompact PSCOMP_1:def 4 :
for T being TopStruct holds
( T is pseudocompact iff for f being RealMap of T st f is continuous holds
f is bounded );

registration
coherence
for b1 being non empty TopSpace st b1 is compact holds
b1 is pseudocompact
proof end;
end;

registration
existence
ex b1 being TopSpace st
( not b1 is empty & b1 is compact )
proof end;
end;

registration
let T be non empty pseudocompact TopSpace;
cluster Function-like V32( the carrier of T, REAL ) continuous -> with_max with_min bounded for Element of K32(K33( the carrier of T,REAL));
coherence
for b1 being RealMap of T st b1 is continuous holds
( b1 is bounded & b1 is with_max & b1 is with_min )
proof end;
end;

theorem Th16: :: PSCOMP_1:16
for T being non empty TopSpace
for X being non empty Subset of T
for Y being compact Subset of T
for f being continuous RealMap of T st X c= Y holds
lower_bound (f | Y) <= lower_bound (f | X)
proof end;

theorem Th17: :: PSCOMP_1:17
for T being non empty TopSpace
for X being non empty Subset of T
for Y being compact Subset of T
for f being continuous RealMap of T st X c= Y holds
upper_bound (f | X) <= upper_bound (f | Y)
proof end;

registration
let n be Element of NAT ;
let X, Y be compact Subset of ();
cluster X /\ Y -> compact for Subset of ();
coherence
for b1 being Subset of () st b1 = X /\ Y holds
b1 is compact
by COMPTS_1:11;
end;

definition
func proj1 -> RealMap of () means :Def5: :: PSCOMP_1:def 5
for p being Point of () holds it . p = p 1 ;
existence
ex b1 being RealMap of () st
for p being Point of () holds b1 . p = p 1
proof end;
uniqueness
for b1, b2 being RealMap of () st ( for p being Point of () holds b1 . p = p 1 ) & ( for p being Point of () holds b2 . p = p 1 ) holds
b1 = b2
proof end;
func proj2 -> RealMap of () means :Def6: :: PSCOMP_1:def 6
for p being Point of () holds it . p = p 2 ;
existence
ex b1 being RealMap of () st
for p being Point of () holds b1 . p = p 2
proof end;
uniqueness
for b1, b2 being RealMap of () st ( for p being Point of () holds b1 . p = p 2 ) & ( for p being Point of () holds b2 . p = p 2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines proj1 PSCOMP_1:def 5 :
for b1 being RealMap of () holds
( b1 = proj1 iff for p being Point of () holds b1 . p = p 1 );

:: deftheorem Def6 defines proj2 PSCOMP_1:def 6 :
for b1 being RealMap of () holds
( b1 = proj2 iff for p being Point of () holds b1 . p = p 2 );

theorem Th18: :: PSCOMP_1:18
for r, s being Real holds proj1 " ].r,s.[ = { |[r1,r2]| where r1, r2 is Real : ( r < r1 & r1 < s ) }
proof end;

theorem Th19: :: PSCOMP_1:19
for P being Subset of ()
for r3, q3 being Real st P = { |[r1,r2]| where r1, r2 is Real : ( r3 < r1 & r1 < q3 ) } holds
P is open
proof end;

theorem Th20: :: PSCOMP_1:20
for r, s being Real holds proj2 " ].r,s.[ = { |[r1,r2]| where r1, r2 is Real : ( r < r2 & r2 < s ) }
proof end;

theorem Th21: :: PSCOMP_1:21
for P being Subset of ()
for r3, q3 being Real st P = { |[r1,r2]| where r1, r2 is Real : ( r3 < r2 & r2 < q3 ) } holds
P is open
proof end;

registration
coherence
proof end;
coherence
proof end;
end;

theorem Th22: :: PSCOMP_1:22
for X being Subset of ()
for p being Point of () st p in X holds
() . p = p 1
proof end;

theorem Th23: :: PSCOMP_1:23
for X being Subset of ()
for p being Point of () st p in X holds
() . p = p 2
proof end;

definition
let X be Subset of ();
func W-bound X -> Real equals :: PSCOMP_1:def 7
lower_bound ();
coherence
lower_bound () is Real
;
func N-bound X -> Real equals :: PSCOMP_1:def 8
upper_bound ();
coherence
upper_bound () is Real
;
func E-bound X -> Real equals :: PSCOMP_1:def 9
upper_bound ();
coherence
upper_bound () is Real
;
func S-bound X -> Real equals :: PSCOMP_1:def 10
lower_bound ();
coherence
lower_bound () is Real
;
end;

:: deftheorem defines W-bound PSCOMP_1:def 7 :
for X being Subset of () holds W-bound X = lower_bound ();

:: deftheorem defines N-bound PSCOMP_1:def 8 :
for X being Subset of () holds N-bound X = upper_bound ();

:: deftheorem defines E-bound PSCOMP_1:def 9 :
for X being Subset of () holds E-bound X = upper_bound ();

:: deftheorem defines S-bound PSCOMP_1:def 10 :
for X being Subset of () holds S-bound X = lower_bound ();

Lm3: for p being Point of ()
for X being non empty compact Subset of () st p in X holds
( lower_bound () <= p 1 & p 1 <= upper_bound () & lower_bound () <= p 2 & p 2 <= upper_bound () )

proof end;

theorem :: PSCOMP_1:24
for p being Point of ()
for X being non empty compact Subset of () st p in X holds
( W-bound X <= p 1 & p 1 <= E-bound X & S-bound X <= p 2 & p 2 <= N-bound X ) by Lm3;

definition
let X be Subset of ();
func SW-corner X -> Point of () equals :: PSCOMP_1:def 11
|[(),()]|;
coherence
|[(),()]| is Point of ()
;
func NW-corner X -> Point of () equals :: PSCOMP_1:def 12
|[(),()]|;
coherence
|[(),()]| is Point of ()
;
func NE-corner X -> Point of () equals :: PSCOMP_1:def 13
|[(),()]|;
coherence
|[(),()]| is Point of ()
;
func SE-corner X -> Point of () equals :: PSCOMP_1:def 14
|[(),()]|;
coherence
|[(),()]| is Point of ()
;
end;

:: deftheorem defines SW-corner PSCOMP_1:def 11 :
for X being Subset of () holds SW-corner X = |[(),()]|;

:: deftheorem defines NW-corner PSCOMP_1:def 12 :
for X being Subset of () holds NW-corner X = |[(),()]|;

:: deftheorem defines NE-corner PSCOMP_1:def 13 :
for X being Subset of () holds NE-corner X = |[(),()]|;

:: deftheorem defines SE-corner PSCOMP_1:def 14 :
for X being Subset of () holds SE-corner X = |[(),()]|;

theorem :: PSCOMP_1:25
for P being Subset of () holds () 1 = () 1
proof end;

theorem :: PSCOMP_1:26
for P being Subset of () holds () 1 = () 1
proof end;

theorem :: PSCOMP_1:27
for P being Subset of () holds () 2 = () 2
proof end;

theorem :: PSCOMP_1:28
for P being Subset of () holds () 2 = () 2
proof end;

definition
let X be Subset of ();
func W-most X -> Subset of () equals :: PSCOMP_1:def 15
(LSeg ((),())) /\ X;
coherence
(LSeg ((),())) /\ X is Subset of ()
;
func N-most X -> Subset of () equals :: PSCOMP_1:def 16
(LSeg ((),())) /\ X;
coherence
(LSeg ((),())) /\ X is Subset of ()
;
func E-most X -> Subset of () equals :: PSCOMP_1:def 17
(LSeg ((),())) /\ X;
coherence
(LSeg ((),())) /\ X is Subset of ()
;
func S-most X -> Subset of () equals :: PSCOMP_1:def 18
(LSeg ((),())) /\ X;
coherence
(LSeg ((),())) /\ X is Subset of ()
;
end;

:: deftheorem defines W-most PSCOMP_1:def 15 :
for X being Subset of () holds W-most X = (LSeg ((),())) /\ X;

:: deftheorem defines N-most PSCOMP_1:def 16 :
for X being Subset of () holds N-most X = (LSeg ((),())) /\ X;

:: deftheorem defines E-most PSCOMP_1:def 17 :
for X being Subset of () holds E-most X = (LSeg ((),())) /\ X;

:: deftheorem defines S-most PSCOMP_1:def 18 :
for X being Subset of () holds S-most X = (LSeg ((),())) /\ X;

registration
let X be non empty compact Subset of ();
cluster W-most X -> non empty compact ;
coherence
( not W-most X is empty & W-most X is compact )
proof end;
cluster N-most X -> non empty compact ;
coherence
( not N-most X is empty & N-most X is compact )
proof end;
cluster E-most X -> non empty compact ;
coherence
( not E-most X is empty & E-most X is compact )
proof end;
cluster S-most X -> non empty compact ;
coherence
( not S-most X is empty & S-most X is compact )
proof end;
end;

definition
let X be Subset of ();
func W-min X -> Point of () equals :: PSCOMP_1:def 19
|[(),(lower_bound (proj2 | ()))]|;
coherence
|[(),(lower_bound (proj2 | ()))]| is Point of ()
;
func W-max X -> Point of () equals :: PSCOMP_1:def 20
|[(),(upper_bound (proj2 | ()))]|;
coherence
|[(),(upper_bound (proj2 | ()))]| is Point of ()
;
func N-min X -> Point of () equals :: PSCOMP_1:def 21
|[(lower_bound (proj1 | ())),()]|;
coherence
|[(lower_bound (proj1 | ())),()]| is Point of ()
;
func N-max X -> Point of () equals :: PSCOMP_1:def 22
|[(upper_bound (proj1 | ())),()]|;
coherence
|[(upper_bound (proj1 | ())),()]| is Point of ()
;
func E-max X -> Point of () equals :: PSCOMP_1:def 23
|[(),(upper_bound (proj2 | ()))]|;
coherence
|[(),(upper_bound (proj2 | ()))]| is Point of ()
;
func E-min X -> Point of () equals :: PSCOMP_1:def 24
|[(),(lower_bound (proj2 | ()))]|;
coherence
|[(),(lower_bound (proj2 | ()))]| is Point of ()
;
func S-max X -> Point of () equals :: PSCOMP_1:def 25
|[(upper_bound (proj1 | ())),()]|;
coherence
|[(upper_bound (proj1 | ())),()]| is Point of ()
;
func S-min X -> Point of () equals :: PSCOMP_1:def 26
|[(lower_bound (proj1 | ())),()]|;
coherence
|[(lower_bound (proj1 | ())),()]| is Point of ()
;
end;

:: deftheorem defines W-min PSCOMP_1:def 19 :
for X being Subset of () holds W-min X = |[(),(lower_bound (proj2 | ()))]|;

:: deftheorem defines W-max PSCOMP_1:def 20 :
for X being Subset of () holds W-max X = |[(),(upper_bound (proj2 | ()))]|;

:: deftheorem defines N-min PSCOMP_1:def 21 :
for X being Subset of () holds N-min X = |[(lower_bound (proj1 | ())),()]|;

:: deftheorem defines N-max PSCOMP_1:def 22 :
for X being Subset of () holds N-max X = |[(upper_bound (proj1 | ())),()]|;

:: deftheorem defines E-max PSCOMP_1:def 23 :
for X being Subset of () holds E-max X = |[(),(upper_bound (proj2 | ()))]|;

:: deftheorem defines E-min PSCOMP_1:def 24 :
for X being Subset of () holds E-min X = |[(),(lower_bound (proj2 | ()))]|;

:: deftheorem defines S-max PSCOMP_1:def 25 :
for X being Subset of () holds S-max X = |[(upper_bound (proj1 | ())),()]|;

:: deftheorem defines S-min PSCOMP_1:def 26 :
for X being Subset of () holds S-min X = |[(lower_bound (proj1 | ())),()]|;

theorem Th29: :: PSCOMP_1:29
for P being Subset of () holds
( () 1 = () 1 & () 1 = () 1 & () 1 = () 1 & () 1 = () 1 & () 1 = () 1 )
proof end;

theorem Th30: :: PSCOMP_1:30
for X being non empty compact Subset of () holds
( () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 )
proof end;

theorem Th31: :: PSCOMP_1:31
for p being Point of ()
for Z being non empty Subset of () st p in W-most Z holds
( p 1 = () 1 & ( Z is compact implies ( () 2 <= p 2 & p 2 <= () 2 ) ) )
proof end;

theorem Th32: :: PSCOMP_1:32
for X being non empty compact Subset of () holds W-most X c= LSeg ((),())
proof end;

theorem :: PSCOMP_1:33
for X being non empty compact Subset of () holds LSeg ((),()) c= LSeg ((),())
proof end;

theorem Th34: :: PSCOMP_1:34
for X being non empty compact Subset of () holds
( W-min X in W-most X & W-max X in W-most X )
proof end;

theorem :: PSCOMP_1:35
for X being non empty compact Subset of () holds
( (LSeg ((),())) /\ X = {()} & (LSeg ((),())) /\ X = {()} )
proof end;

theorem :: PSCOMP_1:36
for X being non empty compact Subset of () st W-min X = W-max X holds
W-most X = {()}
proof end;

theorem Th37: :: PSCOMP_1:37
for P being Subset of () holds
( () 2 = () 2 & () 2 = () 2 & () 2 = () 2 & () 2 = () 2 & () 2 = () 2 )
proof end;

theorem Th38: :: PSCOMP_1:38
for X being non empty compact Subset of () holds
( () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 )
proof end;

theorem Th39: :: PSCOMP_1:39
for p being Point of ()
for Z being non empty Subset of () st p in N-most Z holds
( p 2 = () 2 & ( Z is compact implies ( () 1 <= p 1 & p 1 <= () 1 ) ) )
proof end;

theorem Th40: :: PSCOMP_1:40
for X being non empty compact Subset of () holds N-most X c= LSeg ((),())
proof end;

theorem :: PSCOMP_1:41
for X being non empty compact Subset of () holds LSeg ((),()) c= LSeg ((),())
proof end;

theorem Th42: :: PSCOMP_1:42
for X being non empty compact Subset of () holds
( N-min X in N-most X & N-max X in N-most X )
proof end;

theorem :: PSCOMP_1:43
for X being non empty compact Subset of () holds
( (LSeg ((),())) /\ X = {()} & (LSeg ((),())) /\ X = {()} )
proof end;

theorem :: PSCOMP_1:44
for X being non empty compact Subset of () st N-min X = N-max X holds
N-most X = {()}
proof end;

theorem Th45: :: PSCOMP_1:45
for P being Subset of () holds
( () 1 = () 1 & () 1 = () 1 & () 1 = () 1 & () 1 = () 1 & () 1 = () 1 )
proof end;

theorem Th46: :: PSCOMP_1:46
for X being non empty compact Subset of () holds
( () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 & () 2 <= () 2 )
proof end;

theorem Th47: :: PSCOMP_1:47
for p being Point of ()
for Z being non empty Subset of () st p in E-most Z holds
( p 1 = () 1 & ( Z is compact implies ( () 2 <= p 2 & p 2 <= () 2 ) ) )
proof end;

theorem Th48: :: PSCOMP_1:48
for X being non empty compact Subset of () holds E-most X c= LSeg ((),())
proof end;

theorem :: PSCOMP_1:49
for X being non empty compact Subset of () holds LSeg ((),()) c= LSeg ((),())
proof end;

theorem Th50: :: PSCOMP_1:50
for X being non empty compact Subset of () holds
( E-min X in E-most X & E-max X in E-most X )
proof end;

theorem :: PSCOMP_1:51
for X being non empty compact Subset of () holds
( (LSeg ((),())) /\ X = {()} & (LSeg ((),())) /\ X = {()} )
proof end;

theorem :: PSCOMP_1:52
for X being non empty compact Subset of () st E-min X = E-max X holds
E-most X = {()}
proof end;

theorem Th53: :: PSCOMP_1:53
for P being Subset of () holds
( () 2 = () 2 & () 2 = () 2 & () 2 = () 2 & () 2 = () 2 & () 2 = () 2 )
proof end;

theorem Th54: :: PSCOMP_1:54
for X being non empty compact Subset of () holds
( () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 & () 1 <= () 1 )
proof end;

theorem Th55: :: PSCOMP_1:55
for p being Point of ()
for Z being non empty Subset of () st p in S-most Z holds
( p 2 = () 2 & ( Z is compact implies ( () 1 <= p 1 & p 1 <= () 1 ) ) )
proof end;

theorem Th56: :: PSCOMP_1:56
for X being non empty compact Subset of () holds S-most X c= LSeg ((),())
proof end;

theorem :: PSCOMP_1:57
for X being non empty compact Subset of () holds LSeg ((),()) c= LSeg ((),())
proof end;

theorem Th58: :: PSCOMP_1:58
for X being non empty compact Subset of () holds
( S-min X in S-most X & S-max X in S-most X )
proof end;

theorem :: PSCOMP_1:59
for X being non empty compact Subset of () holds
( (LSeg ((),())) /\ X = {()} & (LSeg ((),())) /\ X = {()} )
proof end;

theorem :: PSCOMP_1:60
for X being non empty compact Subset of () st S-min X = S-max X holds
S-most X = {()}
proof end;

:: Degenerate cases
theorem :: PSCOMP_1:61
for P being Subset of () st W-max P = N-min P holds
W-max P = NW-corner P
proof end;

theorem :: PSCOMP_1:62
for P being Subset of () st N-max P = E-max P holds
N-max P = NE-corner P
proof end;

theorem :: PSCOMP_1:63
for P being Subset of () st E-min P = S-max P holds
E-min P = SE-corner P
proof end;

theorem :: PSCOMP_1:64
for P being Subset of () st S-min P = W-min P holds
S-min P = SW-corner P
proof end;

theorem :: PSCOMP_1:65
for r, s being Real holds
( proj2 . |[r,s]| = s & proj1 . |[r,s]| = r )
proof end;

:: Moved from JORDAN1E, AK, 23.02.2006
theorem :: PSCOMP_1:66
for X being non empty Subset of ()
for Y being compact Subset of () st X c= Y holds
N-bound X <= N-bound Y by Th17;

theorem :: PSCOMP_1:67
for X being non empty Subset of ()
for Y being compact Subset of () st X c= Y holds
E-bound X <= E-bound Y by Th17;

theorem :: PSCOMP_1:68
for X being non empty Subset of ()
for Y being compact Subset of () st X c= Y holds
S-bound X >= S-bound Y by Th16;

theorem :: PSCOMP_1:69
for X being non empty Subset of ()
for Y being compact Subset of () st X c= Y holds
W-bound X >= W-bound Y by Th16;