:: Intermediate Value Theorem and Thickness of Simple Closed Curves
:: by Yatsuka Nakamura and Andrzej Trybulec
::
:: Received November 13, 1997
:: Copyright (c) 1997-2021 Association of Mizar Users

Lm1: for X, Y, Z being non empty TopSpace
for f being continuous Function of X,Y
for g being continuous Function of Y,Z holds g * f is continuous Function of X,Z
;

theorem Th1: :: TOPREAL5:1

for X being non empty TopSpace
for A, B1, B2 being Subset of X st B1 is open & B2 is open & B1 meets A & B2 meets A & A c= B1 \/ B2 & B1 misses B2 holds
not A is connected

proof end;

theorem Th2: :: TOPREAL5:2

for ra, rb being Real st ra <= rb holds
[#] (Closed-Interval-TSpace (ra,rb)) is connected

proof end;

theorem Th3: :: TOPREAL5:3

for A being Subset of R^1
for a being Real st not a in A & ex b, d being Real st
( b in A & d in A & b < a & a < d ) holds
not A is connected

proof end;

:: Intermediate Value Theorem

theorem :: TOPREAL5:4

for X being non empty TopSpace
for xa, xb being Point of X
for a, b, d being Real
for f being continuous Function of X,R^1 st X is connected & f . xa = a & f . xb = b & a <= d & d <= b holds
ex xc being Point of X st f . xc = d

proof end;

theorem :: TOPREAL5:5

for X being non empty TopSpace
for xa, xb being Point of X
for B being Subset of X
for a, b, d being Real
for f being continuous Function of X,R^1 st B is connected & f . xa = a & f . xb = b & a <= d & d <= b & xa in B & xb in B holds
ex xc being Point of X st
( xc in B & f . xc = d )

proof end;

::Intermediate Value Theorem <

theorem Th6: :: TOPREAL5:6

for ra, rb, a, b being Real st ra < rb holds
for f being continuous Function of (Closed-Interval-TSpace (ra,rb)),R^1
for d being Real st f . ra = a & f . rb = b & a < d & d < b holds
ex rc being Real st
( f . rc = d & ra < rc & rc < rb )

proof end;

theorem Th7: :: TOPREAL5:7

for ra, rb, a, b being Real st ra < rb holds
for f being continuous Function of (Closed-Interval-TSpace (ra,rb)),R^1
for d being Real st f . ra = a & f . rb = b & a > d & d > b holds
ex rc being Real st
( f . rc = d & ra < rc & rc < rb )

proof end;

:: Bolzano theorem (intermediate value)

theorem :: TOPREAL5:8

for ra, rb being Real
for g being continuous Function of (Closed-Interval-TSpace (ra,rb)),R^1
for s1, s2 being Real st ra < rb & s1 * s2 < 0 & s1 = g . ra & s2 = g . rb holds
ex r1 being Real st
( g . r1 = 0 & ra < r1 & r1 < rb )

proof end;

theorem Th9: :: TOPREAL5:9

for g being Function of I[01],R^1
for s1, s2 being Real st g is continuous & g . 0 <> g . 1 & s1 = g . 0 & s2 = g . 1 holds
ex r1 being Real st
( 0 < r1 & r1 < 1 & g . r1 = (s1 + s2) / 2 )

proof end;

theorem Th10: :: TOPREAL5:10

for f being Function of (TOP-REAL 2),R^1 st f = proj1 holds
f is continuous

proof end;

theorem Th11: :: TOPREAL5:11

for f being Function of (TOP-REAL 2),R^1 st f = proj2 holds
f is continuous

proof end;

theorem Th12: :: TOPREAL5:12

for P being non empty Subset of (TOP-REAL 2)
for f being Function of I[01],((TOP-REAL 2) | P) st f is continuous holds
ex g being Function of I[01],R^1 st
( g is continuous & ( for r being Real
for q being Point of (TOP-REAL 2) st r in the carrier of I[01] & q = f . r holds
q `1 = g . r ) )

proof end;

theorem Th13: :: TOPREAL5:13

for P being non empty Subset of (TOP-REAL 2)
for f being Function of I[01],((TOP-REAL 2) | P) st f is continuous holds
ex g being Function of I[01],R^1 st
( g is continuous & ( for r being Real
for q being Point of (TOP-REAL 2) st r in the carrier of I[01] & q = f . r holds
q `2 = g . r ) )

proof end;

theorem Th14: :: TOPREAL5:14

for P being non empty Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
for r being Real ex p being Point of (TOP-REAL 2) st
( p in P & not p `2 = r )

proof end;

theorem Th15: :: TOPREAL5:15

for P being non empty Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
for r being Real ex p being Point of (TOP-REAL 2) st
( p in P & not p `1 = r )

proof end;

theorem Th16: :: TOPREAL5:16

for C being non empty compact Subset of (TOP-REAL 2) st C is being_simple_closed_curve holds
N-bound C > S-bound C

proof end;

theorem Th17: :: TOPREAL5:17

for C being non empty compact Subset of (TOP-REAL 2) st C is being_simple_closed_curve holds
E-bound C > W-bound C

proof end;

theorem :: TOPREAL5:18

for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
S-min P <> N-max P

proof end;

theorem :: TOPREAL5:19

for P being non empty compact Subset of (TOP-REAL 2) st P is being_simple_closed_curve holds
W-min P <> E-max P

proof end;

:: Moved from JORDAN1B, AK, 23.02.2006

registration

cluster being_simple_closed_curve -> non horizontal non vertical for Element of K16( the carrier of (TOP-REAL 2));

coherence
for b₁ being Simple_closed_curve holds
( not b₁ is vertical & not b₁ is horizontal )

proof end;

end;