begin
set R = the carrier of R^1;
Lm1:
the carrier of [:R^1,R^1:] = [: the carrier of R^1, the carrier of R^1:]
by BORSUK_1:def 5;
theorem
theorem Th2:
theorem Th3:
theorem
theorem Th5:
theorem Th6:
theorem
begin
registration
let r be
real number ;
let s be
real positive number ;
cluster K155(
r,
(r + s))
-> non
empty ;
coherence
not ].r,(r + s).[ is empty
cluster K153(
r,
(r + s))
-> non
empty ;
coherence
not [.r,(r + s).[ is empty
cluster K154(
r,
(r + s))
-> non
empty ;
coherence
not ].r,(r + s).] is empty
cluster K152(
r,
(r + s))
-> non
empty ;
coherence
not [.r,(r + s).] is empty
cluster K155(
(r - s),
r)
-> non
empty ;
coherence
not ].(r - s),r.[ is empty
cluster K153(
(r - s),
r)
-> non
empty ;
coherence
not [.(r - s),r.[ is empty
cluster K154(
(r - s),
r)
-> non
empty ;
coherence
not ].(r - s),r.] is empty
cluster K152(
(r - s),
r)
-> non
empty ;
coherence
not [.(r - s),r.] is empty
end;
begin
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
theorem
theorem Th24:
theorem Th25:
begin
theorem Th26:
theorem Th27:
theorem
theorem
theorem
theorem
theorem
theorem
theorem Th34:
theorem
theorem
theorem
begin
theorem
theorem
theorem Th40:
theorem Th41:
theorem Th42:
theorem
theorem
theorem
theorem
theorem
theorem Th48:
begin
theorem
theorem
theorem Th51:
for
a,
b,
r,
s being
real number holds
closed_inside_of_rectangle (
a,
b,
r,
s)
= product ((1,2) --> ([.a,b.],[.r,s.]))
theorem Th52:
definition
let a,
b,
c,
d be
real number ;
func Trectangle (
a,
b,
c,
d)
-> SubSpace of
TOP-REAL 2
equals
(TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d));
coherence
(TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d)) is SubSpace of TOP-REAL 2
;
end;
:: deftheorem defines Trectangle TOPREALA:def 1 :
for a, b, c, d being real number holds Trectangle (a,b,c,d) = (TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d));
theorem
canceled;
theorem Th54:
definition
func R2Homeomorphism -> Function of
[:R^1,R^1:],
(TOP-REAL 2) means :
Def2:
for
x,
y being
real number holds
it . [x,y] = <*x,y*>;
existence
ex b1 being Function of [:R^1,R^1:],(TOP-REAL 2) st
for x, y being real number holds b1 . [x,y] = <*x,y*>
uniqueness
for b1, b2 being Function of [:R^1,R^1:],(TOP-REAL 2) st ( for x, y being real number holds b1 . [x,y] = <*x,y*> ) & ( for x, y being real number holds b2 . [x,y] = <*x,y*> ) holds
b1 = b2
end;
:: deftheorem Def2 defines R2Homeomorphism TOPREALA:def 2 :
for b1 being Function of [:R^1,R^1:],(TOP-REAL 2) holds
( b1 = R2Homeomorphism iff for x, y being real number holds b1 . [x,y] = <*x,y*> );
theorem Th55:
theorem Th56:
theorem Th57:
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (r,s)):] is
Function of
[:(Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (r,s)):],
(Trectangle (a,b,r,s))
theorem Th58:
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
h being
Function of
[:(Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (r,s)):],
(Trectangle (a,b,r,s)) st
h = R2Homeomorphism | the
carrier of
[:(Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (r,s)):] holds
h is
being_homeomorphism
theorem
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
[:(Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (r,s)):],
Trectangle (
a,
b,
r,
s)
are_homeomorphic
theorem Th60:
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
Subset of
(Closed-Interval-TSpace (a,b)) for
B being
Subset of
(Closed-Interval-TSpace (r,s)) holds
product ((1,2) --> (A,B)) is
Subset of
(Trectangle (a,b,r,s))
theorem
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
open Subset of
(Closed-Interval-TSpace (a,b)) for
B being
open Subset of
(Closed-Interval-TSpace (r,s)) holds
product ((1,2) --> (A,B)) is
open Subset of
(Trectangle (a,b,r,s))
theorem
for
a,
b,
r,
s being
real number st
a <= b &
r <= s holds
for
A being
closed Subset of
(Closed-Interval-TSpace (a,b)) for
B being
closed Subset of
(Closed-Interval-TSpace (r,s)) holds
product ((1,2) --> (A,B)) is
closed Subset of
(Trectangle (a,b,r,s))