Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997 Association of Mizar Users

### The Ordering of Points on a Curve. Part II

by
Yatsuka Nakamura

MML identifier: JORDAN5C
[ Mizar article, MML identifier index ]

```environ

vocabulary ARYTM_1, EUCLID, PRE_TOPC, BORSUK_1, RELAT_1, TOPREAL1, FUNCT_1,
TOPS_2, SUBSET_1, ORDINAL2, BOOLE, COMPTS_1, JORDAN3, RCOMP_1, FINSEQ_1,
TREAL_1, SEQ_1, TOPMETR, JORDAN5C, FINSEQ_4;
notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, REAL_1, NAT_1, RCOMP_1, RELAT_1,
FINSEQ_1, FUNCT_1, FUNCT_2, TOPMETR, FINSEQ_4, JORDAN3, STRUCT_0,
TOPREAL1, PRE_TOPC, TOPS_2, COMPTS_1, EUCLID, TREAL_1;
constructors REAL_1, NAT_1, TOPS_2, COMPTS_1, RCOMP_1, TREAL_1, JORDAN3,
TOPREAL1, FINSEQ_4, MEMBERED;
clusters BORSUK_1, EUCLID, FUNCT_1, PRE_TOPC, RELSET_1, STRUCT_0, XREAL_0,
ARYTM_3, MEMBERED;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;

begin

theorem :: JORDAN5C:1    :: Characterization of First Point Intersecting Q
for P, Q being Subset of TOP-REAL 2,
p1, p2, q1 being Point of TOP-REAL 2,
f being map of I, (TOP-REAL 2)|P, s1 be Real
st P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1 = q1 & f is_homeomorphism
& f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1 &
(for t being Real st 0 <= t & t < s1 holds not f.t in Q) holds
for g being map of I, (TOP-REAL 2)|P, s2 be Real st
g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = q1
& 0 <= s2 & s2 <= 1 holds
(for t being Real st 0 <= t & t < s2 holds not g.t in Q);

definition
let P, Q be Subset of TOP-REAL 2,
p1, p2 be Point of TOP-REAL 2;
assume  P meets Q & P /\ Q is closed & P is_an_arc_of p1,p2;
func First_Point(P,p1,p2,Q) -> Point of TOP-REAL 2 means
:: JORDAN5C:def 1
it in P /\ Q &
for g being map of I, (TOP-REAL 2)|P, s2 being Real st
g is_homeomorphism & g.0 = p1 & g.1 = p2
& g.s2 = it & 0 <= s2 & s2 <= 1 holds
(for t being Real st 0 <= t & t < s2 holds not g.t in Q);
end;

theorem :: JORDAN5C:2
for P, Q being Subset of TOP-REAL 2,
p, p1, p2 being Point of TOP-REAL 2 st
p in P & P is_an_arc_of p1, p2 & Q = {p} holds
First_Point (P,p1,p2,Q) = p;

theorem :: JORDAN5C:3
for P being Subset of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
p1, p2 being Point of TOP-REAL 2 st
p1 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds
First_Point (P, p1, p2, Q) = p1;

theorem :: JORDAN5C:4    ::Characterization of Last Point Intersecting Q

for P, Q being Subset of TOP-REAL 2,
p1, p2, q1 being Point of TOP-REAL 2,
f being map of I, (TOP-REAL 2)|P,
s1 be Real
st P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1 = q1 &
f is_homeomorphism & f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1 &
(for t being Real st 1 >= t & t > s1 holds not f.t in Q) holds
for g being map of I, (TOP-REAL 2)|P, s2 being Real st
g is_homeomorphism
& g.0 = p1 & g.1 = p2 & g.s2 = q1
& 0 <= s2 & s2 <= 1 holds
(for t being Real st 1 >= t & t > s2 holds not g.t in Q);

definition
let P, Q be Subset of TOP-REAL 2,
p1,p2 be Point of TOP-REAL 2;
assume  P meets Q & P /\ Q is closed & P is_an_arc_of p1,p2;
func Last_Point (P,p1,p2,Q) -> Point of TOP-REAL 2 means
:: JORDAN5C:def 2
it in P /\ Q &
for g being map of I, (TOP-REAL 2)|P, s2 be Real st
g is_homeomorphism
& g.0 = p1 & g.1 = p2
& g.s2 = it & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds not g.t in Q;
end;

theorem :: JORDAN5C:5
for P, Q being Subset of TOP-REAL 2,
p, p1,p2 being Point of TOP-REAL 2 st
p in P & P is_an_arc_of p1, p2 & Q = {p} holds
Last_Point (P, p1, p2, Q) = p;

theorem :: JORDAN5C:6
for P,Q being Subset of TOP-REAL 2,
p1, p2 being Point of TOP-REAL 2 st
p2 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds
Last_Point (P, p1, p2, Q) = p2;

theorem :: JORDAN5C:7
for P being Subset of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
p1, p2 being Point of TOP-REAL 2 st
P c= Q & P is closed & P is_an_arc_of p1, p2 holds
First_Point (P, p1, p2, Q) = p1 &
Last_Point (P, p1, p2, Q) = p2;

begin :: The ordering of points on a curve

definition let P be Subset of TOP-REAL 2,
p1, p2, q1, q2 be Point of TOP-REAL 2;
pred LE q1, q2, P, p1, p2 means
:: JORDAN5C:def 3
q1 in P & q2 in P &
for g being map of I, (TOP-REAL 2)|P,
s1, s2 being Real st
g is_homeomorphism
& g.0 = p1 & g.1 = p2
& g.s1 = q1 & 0 <= s1 & s1 <= 1
& g.s2 = q2 & 0 <= s2 & s2 <= 1
holds s1 <= s2;
end;

theorem :: JORDAN5C:8
for P being Subset of TOP-REAL 2,
p1, p2, q1, q2 being Point of TOP-REAL 2,
g being map of I, (TOP-REAL 2)|P,
s1, s2 being Real st
P is_an_arc_of p1,p2 & g is_homeomorphism & g.0 = p1 & g.1 = p2
& g.s1 = q1 & 0 <= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <=
s2 holds
LE q1,q2,P,p1,p2;

theorem :: JORDAN5C:9
for P being Subset of TOP-REAL 2,
p1,p2,q1 being Point of TOP-REAL 2
st P is_an_arc_of p1,p2 & q1 in P holds
LE q1,q1,P,p1,p2;

theorem :: JORDAN5C:10
for P being Subset of TOP-REAL 2,
p1,p2,q1 being Point of TOP-REAL 2
st P is_an_arc_of p1,p2 & q1 in P holds
LE p1,q1,P,p1,p2 & LE q1,p2,P,p1,p2;

theorem :: JORDAN5C:11
for P being Subset of TOP-REAL 2,
p1,p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 holds
LE p1,p2,P,p1,p2;

theorem :: JORDAN5C:12
for P being Subset of TOP-REAL 2,
p1, p2, q1, q2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 &
LE q1,q2,P,p1,p2 & LE q2,q1,P,p1,p2 holds q1=q2;

theorem :: JORDAN5C:13
for P being Subset of TOP-REAL 2,
p1,p2,q1,q2,q3 being Point of TOP-REAL 2
st P is_an_arc_of p1,p2 &
LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
LE q1,q3,P,p1,p2;

theorem :: JORDAN5C:14
for P being Subset of TOP-REAL 2,
p1, p2, q1, q2 being Point of TOP-REAL 2 st
P is_an_arc_of p1, p2 & q1 in P & q2 in P & q1 <> q2
holds
LE q1,q2,P,p1,p2 & not LE q2,q1,P,p1,p2 or
LE q2,q1,P,p1,p2 & not LE q1,q2,P,p1,p2;

begin :: Some properties of the ordering of points on a curve

theorem :: JORDAN5C:15
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
q being Point of TOP-REAL 2 st
f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q holds
LE First_Point(L~f,f/.1,f/.len f,Q), q, L~f, f/.1, f/.len f;

theorem :: JORDAN5C:16
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
q being Point of TOP-REAL 2 st
f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q holds
LE q, Last_Point(L~f,f/.1,f/.len f,Q), L~f, f/.1, f/.len f;

theorem :: JORDAN5C:17
for q1, q2, p1, p2 being Point of TOP-REAL 2 st p1 <> p2 holds
LE q1, q2, LSeg(p1, p2), p1, p2 implies LE q1, q2, p1, p2;

theorem :: JORDAN5C:18
for P, Q being Subset of TOP-REAL 2,
p1, p2 being Point of TOP-REAL 2 st
P is_an_arc_of p1,p2 & P meets Q & P /\ Q is closed holds
First_Point(P,p1,p2,Q) = Last_Point(P,p2,p1,Q) &
Last_Point(P,p1,p2,Q) = First_Point(P,p2,p1,Q);

theorem :: JORDAN5C:19
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
i being Nat
st L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f &
First_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i) holds
First_Point (L~f, f/.1, f/.len f, Q) =
First_Point (LSeg (f, i), f/.i, f/.(i+1), Q);

theorem :: JORDAN5C:20
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
i being Nat
st L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f &
Last_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i) holds
Last_Point (L~f, f/.1, f/.len f, Q) =
Last_Point (LSeg (f, i), f/.i, f/.(i+1), Q);

theorem :: JORDAN5C:21
for f being FinSequence of TOP-REAL 2,
i be Nat st 1 <= i & i+1 <= len f & f is_S-Seq &
First_Point (L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i) holds
First_Point (L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.i;

theorem :: JORDAN5C:22
for f being FinSequence of TOP-REAL 2,
i be Nat st 1 <= i & i+1 <= len f & f is_S-Seq &
Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i)
holds
Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.(i+1);

theorem :: JORDAN5C:23
for f be FinSequence of TOP-REAL 2,
i be Nat st
f is_S-Seq & 1 <= i & i+1 <= len f holds
LE f/.i, f/.(i+1), L~f, f/.1, f/.len f;

theorem :: JORDAN5C:24
for f be FinSequence of TOP-REAL 2,
i, j be Nat st
f is_S-Seq & 1 <= i & i <= j & j <= len f holds
LE f/.i, f/.j, L~f, f/.1, f/.len f;

theorem :: JORDAN5C:25
for f being FinSequence of TOP-REAL 2,
q being Point of TOP-REAL 2,
i being Nat st
f is_S-Seq & 1 <= i & i+1 <= len f & q in LSeg(f,i) holds
LE f/.i, q, L~f, f/.1, f/.len f;

theorem :: JORDAN5C:26
for f being FinSequence of TOP-REAL 2,
q being Point of TOP-REAL 2,
i being Nat st
f is_S-Seq & 1<=i & i+1<=len f & q in LSeg(f,i) holds
LE q, f/.(i+1), L~f, f/.1, f/.len f;

theorem :: JORDAN5C:27
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
q being Point of TOP-REAL 2,
i, j being Nat st
L~f meets Q & f is_S-Seq & Q is closed &
First_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f &
q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q &
First_Point (L~f,f/.1,f/.len f,Q) <> q
holds
i <= j &
(i=j implies LE First_Point(L~f,f/.1,f/.len f,Q), q, f/.i, f/.(i+1));

theorem :: JORDAN5C:28
for f being FinSequence of TOP-REAL 2,
Q being Subset of TOP-REAL 2,
q being Point of TOP-REAL 2,
i, j being Nat st
L~f meets Q & f is_S-Seq & Q is closed &
Last_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f &
q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q &
Last_Point (L~f,f/.1,f/.len f,Q) <> q holds
i >= j &
(i=j implies LE q, Last_Point(L~f,f/.1,f/.len f,Q), f/.i, f/.(i+1));

theorem :: JORDAN5C:29
for f being FinSequence of TOP-REAL 2,
q1, q2 being Point of TOP-REAL 2,
i being Nat st
q1 in LSeg(f,i) & q2 in LSeg(f,i) & f is_S-Seq & 1 <= i & i + 1 <= len f
holds LE q1, q2, L~f, f/.1, f/.len f implies
LE q1, q2, LSeg (f,i), f/.i, f/.(i+1);

theorem :: JORDAN5C:30
for f being FinSequence of TOP-REAL 2,
q1, q2 being Point of TOP-REAL 2 st
q1 in L~f & q2 in L~f & f is_S-Seq & q1 <> q2
holds LE q1, q2, L~f, f/.1, f/.len f iff
for i, j being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,j)
& 1 <= i & i+1 <= len f
& 1 <= j & j+1 <= len f holds
i <= j & (i = j implies LE q1,q2,f/.i,f/.(i+1));

```