JOURNAL OF FORMALIZED MATHEMATICS
Volume 6, Released 1994, Published 1999
Institute of Mathematics, University of Bialystok
Extremal Properties of Vertices on Special
Polygons, Part I
Yatsuka Nakamura
Shinshu University
Nagano
Czeslaw Bylinski
Warsaw University
Bialystok
Summary. First, extremal properties of endpoints of line segments in ndimensional Euclidean space are discussed. Some topological properties of line segments are also discussed. Secondly, extremal properties of vertices of special polygons
which consist of horizontal and vertical line segments in 2-dimensional Euclidean
space, are also derived.
MML Identier: SPPOL_1.
WWW: http://mizar.org/JFM/Vol6/sppol_1.html
The articles [20], [25], [2], [13], [24], [6], [19], [22], [17], [16], [7], [18], [21], [10], [1], [3], [14],
[5], [9], [4], [8], [23], [11], [15], and [12] provide the notation and terminology for this paper.
1. Preliminaries
The following propositions are true:
(1) For every nite sequence f holds f is trivial i len f < 2:
(2) For every nite set A holds A is trivial i card A < 2:
(3) For every set A holds A is non trivial i there exist sets a1 , a2 such that a1 2 A
and a2 2 A and a1 6= a2 :
(4) Let D be a non empty set and A be a subset of D. Then A is non trivial if and only
if there exist elements d1 , d2 of D such that d1 2 A and d2 2 A and d1 6= d2 :
We adopt the following rules: n, i, k, m denote natural numbers and r, r1 , r2 , s, s1 , s2
denote real numbers.
The following propositions are true:
(5) If n k; then n ? 1 k and n ? 1 < k and n k + 1 and n < k + 1:
(6) If n < k; then n ? 1 k and n ? 1 < k and n + 1 k and n k ? 1 and n k + 1
and n < k + 1:
(7) If 1 k ? m and k ? m n; then k ? m 2 Seg n and k ? m is a natural number.
(8) If r1 0 and r2 0 and r1 + r2 = 0; then r1 = 0 and r2 = 0:
(9) If r1 0 and r2 0 and r1 + r2 = 0; then r1 = 0 and r2 = 0:
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extremal properties of vertices on special . . .
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(10) If 0 r1 and r1 1 and 0 r2 and r2 1 and r1 r2 = 1; then r1 = 1 and r2 = 1:
(11) If r1 0 and r2 0 and s1 0 and s2 0 and r1 s1 + r2 s2 = 0; then r1 = 0 or
s1 = 0 but r2 = 0 or s2 = 0:
(12) If 0 r and r 1 and s1 0 and s2 0 and r s1 + (1 ? r) s2 = 0; then r = 0
and s2 = 0 or r = 1 and s1 = 0 or s1 = 0 and s2 = 0:
(13) If r < r1 and r < r2 ; then r < min(r1 ; r2 ):
(14) If r > r1 and r > r2 ; then r > max(r1 ; r2 ):
In this article we present several logical schemes. The scheme FinSeqFam deals with a
non empty set A; a nite sequence B of elements of A; a binary functor F yielding a set,
and a unary predicate P ; and states that:
fF (B; i) : i 2 dom B ^ P [i]g is nite
for all values of the parameters.
The scheme FinSeqFam' concerns a non empty set A; a nite sequence B of elements of
A; a binary functor F yielding a set, and a unary predicate P ; and states that:
fF (B; i) : 1 i ^ i len B ^ P [i]g is nite
for all values of the parameters.
One can prove the following propositions:
(15)
(16)
(17)
(18)
(19)
For all elements x1 , x2 , x3 of Rn holds jx1 ? x2 j ? jx2 ? x3 j jx1 ? x3 j:
For all elements x1 , x2 , x3 of Rn holds jx2 ? x1 j ? jx2 ? x3 j jx3 ? x1 j:
Every point of ETn is an element of Rn and a point of E n .
Every point of E n is an element of Rn and a point of ETn .
Every element of Rn is a point of E n and a point of ETn .
2. Properties of line segments
In the sequel p, p1 , p2 , q1 , q2 are points of ETn .
Next we state a number of propositions:
(20) For all points u1 , u2 of E n and for all elements v1 , v2 of Rn such that v1 = u1 and
v2 = u2 holds (u1 ; u2 ) = jv1 ? v2 j:
(21) For all p, p1 , p2 such that p 2 L(p1 ; p2 ) there exists r such that 0 r and r 1
and p = (1 ? r) p1 + r p2 :
(22) For all p1 , p2 , r such that 0 r and r 1 holds (1 ? r) p1 + r p2 2 L(p1 ; p2 ):
(23) Let P be a non empty subset of ETn . Suppose P is closed and P L(p1 ; p2 ): Then
there exists s such that (1 ? s) p1 + s p2 2 P and for every r such that 0 r and
r 1 and (1 ? r) p1 + r p2 2 P holds s r:
(24) For all p1 , p2 , q1 , q2 such that L(q1 ; q2 ) L(p1 ; p2 ) and p1 2 L(q1 ; q2 ) holds p1 = q1
or p1 = q2 :
(25) For all p1 , p2 , q1 , q2 such that L(p1 ; p2) = L(q1 ; q2 ) holds p1 = q1 and p2 = q2 or
p1 = q2 and p2 = q1 :
(26) ETn is a T2 space.
(27) fpg is closed.
(28) L(p1 ; p2 ) is compact.
extremal properties of vertices on special . . .
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(29) L(p1 ; p2 ) is closed.
Let us consider n, p and let P be a subset of the carrier of ETn . We say that p is extremal
in P if and only if:
(Def. 1) p 2 P and for all p1 , p2 such that p 2 L(p1 ; p2 ) and L(p1 ; p2 ) P holds p = p1 or
p = p2 :
The following propositions are true:
(30) For all subsets P , Q of the carrier of ETn such that p is extremal in P and Q P
and p 2 Q holds p is extremal in Q.
(31) p is extremal in fpg.
(32) p1 is extremal in L(p1 ; p2 ).
(33) p2 is extremal in L(p1 ; p2 ).
(34) If p is extremal in L(p1 ; p2 ), then p = p1 or p = p2 :
3. Alternating special sequences
We use the following convention: P , Q denote subsets of ET2 , f , f1 , f2 denote nite sequences
of elements of the carrier of ET2 , and p, p1 , p2 , p3 , q denote points of ET2 .
We now state the proposition
(35) For all p1 , p2 such that (p1 )1 6= (p2 )1 and (p1 )2 6= (p2 )2 there exists p such that
p 2 L(p1 ; p2 ) and p1 6= (p1 )1 and p1 6= (p2 )1 and p2 6= (p1 )2 and p2 6= (p2 )2 :
Let us consider P . We say that P is horizontal if and only if:
(Def. 2) For all p, q such that p 2 P and q 2 P holds p2 = q2 :
We say that P is vertical if and only if:
(Def. 3) For all p, q such that p 2 P and q 2 P holds p1 = q1 :
Let us mention that every subset of ET2 which is non trivial and horizontal is also non
vertical and every subset of ET2 which is non trivial and vertical is also non horizontal.
We now state a number of propositions:
(36) p2 = q2 i L(p; q) is horizontal.
(37) p1 = q1 i L(p; q) is vertical.
(38) If p1 2 L(p; q) and p2 2 L(p; q) and (p1 )1 6= (p2 )1 and (p1 )2 = (p2 )2 ; then L(p; q) is
horizontal.
(39) If p1 2 L(p; q) and p2 2 L(p; q) and (p1 )2 6= (p2 )2 and (p1 )1 = (p2 )1 ; then L(p; q) is
vertical.
(40) L(f; i) is closed.
(41) If f is special, then L(f; i) is vertical or L(f; i) is horizontal.
(42) If f is one-to-one and 1 i and i + 1 len f; then L(f; i) is non trivial.
(43) If f is one-to-one and 1 i and i + 1 len f and L(f; i) is vertical, then L(f; i) is
non horizontal.
(44) For every f holds fL(f; i) : 1 i ^ i len f g is nite.
(45) For every f holds fL(f; i) : 1 i ^ i + 1 len f g is nite.
extremal properties of vertices on special . . .
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(46) For every f holds fL(f; i) : 1 i ^ i len f g is a family of subsets of ET2 .
(47) For every f holds fL(f; i) : 1 i ^ i + 1 len f g is a family of subsets of ET2 .
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(48) For every f such that Q = fL(f; i) : 1 i ^ i + 1 len f g holds Q is closed.
(49) Le(f ) is closed.
Let I1 be a nite sequence of elements of ET2 . We say that I1 is alternating if and only
if:
(Def. 4) For every i such that 1 i and i + 2 len I1 holds (i I1 )1 6= (i+2 I1 )1 and
(i I1 )2 6= (i+2 I1 )2 :
Next we state a number of propositions:
(50) If f is special and alternating and 1 i and i + 2 len f and (i f )1 = (i+1 f )1 ;
then (i+1 f )2 = (i+2 f )2 :
(51) If f is special and alternating and 1 i and i + 2 len f and (i f )2 = (i+1 f )2 ;
then (i+1 f )1 = (i+2 f )1 :
(52) Suppose f is special and alternating and 1 i and i + 2 len f and p1 = i f and
p2 = i+1 f and p3 = i+2 f: Then (p1 )1 = (p2 )1 and (p3 )1 6= (p2 )1 or (p1 )2 = (p2 )2
and (p3 )2 6= (p2 )2 :
(53) Suppose f is special and alternating and 1 i and i + 2 len f and p1 = i f and
p2 = i+1 f and p3 = i+2 f: Then (p2 )1 = (p3 )1 and (p1 )1 6= (p2 )1 or (p2 )2 = (p3 )2
and (p1 )2 6= (p2 )2 :
(54) If f is special and alternating and 1 i and i + 2 len f; then L(i f; i+2 f ) 6
L(f; i) [ L(f; i + 1):
(55) If f is special and alternating and 1 i and i + 2 len f and L(f; i) is vertical,
then L(f; i + 1) is horizontal.
(56) If f is special and alternating and 1 i and i + 2 len f and L(f; i) is horizontal,
then L(f; i + 1) is vertical.
(57) Suppose f is special and alternating and 1 i and i + 2 len f: Then L(f; i) is
vertical and L(f; i + 1) is horizontal or L(f; i) is horizontal and L(f; i + 1) is vertical.
(58) Suppose f is special and alternating and 1 i and i +2 len f and i+1 f 2 L(p; q)
and L(p; q) L(f; i) [ L(f; i + 1): Then i+1 f = p or i+1 f = q:
(59) If f is special and alternating and 1 i and i + 2 len f; then i+1 f is extremal
in L(f; i) [ L(f; i + 1):
(60) Let u be a point of E 2 . Suppose f is special and alternating and 1 i and i + 2 len f and u = i+1 f and i+1 f 2 L(p; q) and i+1 f 6= q and p 2= L(f; i) [ L(f; i + 1):
Let given s. If s > 0; then there exists p3 such that p3 2= L(f; i) [ L(f; i + 1) and
p3 2 L(p; q) and p3 2 Ball(u; s):
Let us consider f1 , f2 , P . We say that f1 and f2 are generators of P if and only if the
conditions (Def. 5) are satised.
(Def. 5)(i) f1 is alternating and special sequence,
(ii) f2 is alternating and special sequence,
(iii) 1 f1 = 1 f2 ;
(iv) len f1 f1 = len f2 f2 ;
(v) h2 f1; 1 f1 ; 2 f2 i is alternating,
extremal properties of vertices on special . . .
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(vi) hlen f1 ?1 f1; len f1 f1 ; len f2 ?1 f2 i is alternating,
(vii) 1 f1 6= len f1 f1 ;
(viii) Le(f1 ) \ Le(f2 ) = f1 f1 ; len f1 f1g; and
(ix) P = Le(f1 ) [ Le(f2 ):
One can prove the following proposition
(61) If f1 and f2 are generators of P and 1 < i and i < len f1 ; then i f1 is extremal in
P.
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Received May 11, 1994
Published May 12, 1999