:: Basic Properties of Connecting Points with Line Segments in ${\calE}^2_{\rm T}$
:: by Yatsuka Nakamura and Jaros{\l}aw Kotowicz
::
:: Received August 24, 1992
:: Copyright (c) 1992-2012 Association of Mizar Users


begin

begin

theorem Th1: :: TOPREAL3:1
for x, y, z being set holds
( 1 in dom <*x,y,z*> &amp; 2 in dom <*x,y,z*> &amp; 3 in dom <*x,y,z*> )
proofend;

theorem Th2: :: TOPREAL3:2
for p1, p2 being Point of (TOP-REAL 2) holds
( (p1 + p2) `1 = (p1 `1) + (p2 `1) &amp; (p1 + p2) `2 = (p1 `2) + (p2 `2) )
proofend;

theorem :: TOPREAL3:3
for p1, p2 being Point of (TOP-REAL 2) holds
( (p1 - p2) `1 = (p1 `1) - (p2 `1) &amp; (p1 - p2) `2 = (p1 `2) - (p2 `2) )
proofend;

theorem Th4: :: TOPREAL3:4
for p being Point of (TOP-REAL 2)
for r being real number holds
( (r * p) `1 = r * (p `1) &amp; (r * p) `2 = r * (p `2) )
proofend;

theorem Th5: :: TOPREAL3:5
for p1, p2 being Point of (TOP-REAL 2)
for r1, s1, r2, s2 being real number st p1 = <*r1,s1*> &amp; p2 = <*r2,s2*> holds
( p1 + p2 = <*(r1 + r2),(s1 + s2)*> &amp; p1 - p2 = <*(r1 - r2),(s1 - s2)*> )
proofend;

theorem Th6: :: TOPREAL3:6
for p, q being Point of (TOP-REAL 2) st p `1 = q `1 &amp; p `2 = q `2 holds
p = q
proofend;

theorem Th7: :: TOPREAL3:7
for p1, p2 being Point of (TOP-REAL 2)
for u1, u2 being Point of (Euclid 2) st u1 = p1 &amp; u2 = p2 holds
(Pitag_dist 2) . (u1,u2) = sqrt ((((p1 `1) - (p2 `1)) ^2) + (((p1 `2) - (p2 `2)) ^2))
proofend;

theorem Th8: :: TOPREAL3:8
for n being Nat holds the carrier of (TOP-REAL n) = the carrier of (Euclid n) by EUCLID:22;

theorem Th9: :: TOPREAL3:9
for r1, s1, r being real number st r1 <= s1 holds
{  p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = r &amp; r1 <= p1 `2 &amp; p1 `2 <= s1 )  }  = LSeg (|[r,r1]|,|[r,s1]|)
proofend;

theorem Th10: :: TOPREAL3:10
for r1, s1, r being real number st r1 <= s1 holds
{  p1 where p1 is Point of (TOP-REAL 2) : ( p1 `2 = r &amp; r1 <= p1 `1 &amp; p1 `1 <= s1 )  }  = LSeg (|[r1,r]|,|[s1,r]|)
proofend;

theorem :: TOPREAL3:11
for p being Point of (TOP-REAL 2)
for r, r1, s1 being real number st p in LSeg (|[r,r1]|,|[r,s1]|) holds
p `1 = r
proofend;

theorem :: TOPREAL3:12
for p being Point of (TOP-REAL 2)
for r1, r, s1 being real number st p in LSeg (|[r1,r]|,|[s1,r]|) holds
p `2 = r
proofend;

theorem :: TOPREAL3:13
for p, q being Point of (TOP-REAL 2) st p `1 <> q `1 &amp; p `2 = q `2 holds
|[(((p `1) + (q `1)) / 2),(p `2)]| in LSeg (p,q)
proofend;

theorem :: TOPREAL3:14
for p, q being Point of (TOP-REAL 2) st p `1 = q `1 &amp; p `2 <> q `2 holds
|[(p `1),(((p `2) + (q `2)) / 2)]| in LSeg (p,q)
proofend;

theorem Th15: :: TOPREAL3:15
for p, p1, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2)
for i, j being Nat st f = <*p,p1,q*> &amp; i <> 0 &amp; j > i + 1 holds
LSeg (f,j) = {}
proofend;

theorem :: TOPREAL3:16
for p1, p2, p3 being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st f = <*p1,p2,p3*> holds
L~ f = (LSeg (p1,p2)) \/ (LSeg (p2,p3))
proofend;

theorem Th17: :: TOPREAL3:17
for f being FinSequence of (TOP-REAL 2)
for j, i being Element of NAT st j in dom (f | i) &amp; j + 1 in dom (f | i) holds
LSeg (f,j) = LSeg ((f | i),j)
proofend;

theorem :: TOPREAL3:18
for f, h being FinSequence of (TOP-REAL 2)
for j being Element of NAT st j in dom f &amp; j + 1 in dom f holds
LSeg ((f ^ h),j) = LSeg (f,j)
proofend;

theorem Th19: :: TOPREAL3:19
for n being Element of NAT
for f being FinSequence of (TOP-REAL n)
for i being Nat holds LSeg (f,i) c= L~ f
proofend;

theorem :: TOPREAL3:20
for f being FinSequence of (TOP-REAL 2)
for i being Element of NAT holds L~ (f | i) c= L~ f
proofend;

theorem Th21: :: TOPREAL3:21
for r being real number
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p1 in Ball (u,r) &amp; p2 in Ball (u,r) holds
LSeg (p1,p2) c= Ball (u,r)
proofend;

theorem :: TOPREAL3:22
for p1, p2, p being Point of (TOP-REAL 2)
for r1, s1, r2, s2, r being real number
for u being Point of (Euclid 2) st u = p1 &amp; p1 = |[r1,s1]| &amp; p2 = |[r2,s2]| &amp; p = |[r2,s1]| &amp; p2 in Ball (u,r) holds
p in Ball (u,r)
proofend;

theorem :: TOPREAL3:23
for s, r1, r, s1 being real number
for u being Point of (Euclid 2) st |[s,r1]| in Ball (u,r) &amp; |[s,s1]| in Ball (u,r) holds
|[s,((r1 + s1) / 2)]| in Ball (u,r)
proofend;

theorem :: TOPREAL3:24
for r1, s, r, s1 being real number
for u being Point of (Euclid 2) st |[r1,s]| in Ball (u,r) &amp; |[s1,s]| in Ball (u,r) holds
|[((r1 + s1) / 2),s]| in Ball (u,r)
proofend;

theorem :: TOPREAL3:25
for r1, r2, r, s1, s2 being real number
for u being Point of (Euclid 2) st |[r1,r2]| in Ball (u,r) &amp; |[s1,s2]| in Ball (u,r) &amp; not |[r1,s2]| in Ball (u,r) holds
|[s1,r2]| in Ball (u,r)
proofend;

theorem :: TOPREAL3:26
for f being FinSequence of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2)
for m being Element of NAT st not f /. 1 in Ball (u,r) &amp; 1 <= m &amp; m <= (len f) - 1 &amp; ( for i being Element of NAT st 1 <= i &amp; i <= (len f) - 1 &amp; (LSeg (f,i)) /\ (Ball (u,r)) <> {} holds
m <= i ) holds
not f /. m in Ball (u,r)
proofend;

theorem :: TOPREAL3:27
for q, p2, p being Point of (TOP-REAL 2) st q `2 = p2 `2 &amp; p `2 <> p2 `2 holds
((LSeg (p2,|[(p2 `1),(p `2)]|)) \/ (LSeg (|[(p2 `1),(p `2)]|,p))) /\ (LSeg (q,p2)) = {p2}
proofend;

theorem :: TOPREAL3:28
for q, p2, p being Point of (TOP-REAL 2) st q `1 = p2 `1 &amp; p `1 <> p2 `1 holds
((LSeg (p2,|[(p `1),(p2 `2)]|)) \/ (LSeg (|[(p `1),(p2 `2)]|,p))) /\ (LSeg (q,p2)) = {p2}
proofend;

theorem Th29: :: TOPREAL3:29
for p, q being Point of (TOP-REAL 2) holds (LSeg (p,|[(p `1),(q `2)]|)) /\ (LSeg (|[(p `1),(q `2)]|,q)) = {|[(p `1),(q `2)]|}
proofend;

theorem Th30: :: TOPREAL3:30
for p, q being Point of (TOP-REAL 2) holds (LSeg (p,|[(q `1),(p `2)]|)) /\ (LSeg (|[(q `1),(p `2)]|,q)) = {|[(q `1),(p `2)]|}
proofend;

theorem Th31: :: TOPREAL3:31
for p, q being Point of (TOP-REAL 2) st p `1 = q `1 &amp; p `2 <> q `2 holds
(LSeg (p,|[(p `1),(((p `2) + (q `2)) / 2)]|)) /\ (LSeg (|[(p `1),(((p `2) + (q `2)) / 2)]|,q)) = {|[(p `1),(((p `2) + (q `2)) / 2)]|}
proofend;

theorem Th32: :: TOPREAL3:32
for p, q being Point of (TOP-REAL 2) st p `1 <> q `1 &amp; p `2 = q `2 holds
(LSeg (p,|[(((p `1) + (q `1)) / 2),(p `2)]|)) /\ (LSeg (|[(((p `1) + (q `1)) / 2),(p `2)]|,q)) = {|[(((p `1) + (q `1)) / 2),(p `2)]|}
proofend;

theorem :: TOPREAL3:33
for f being FinSequence of (TOP-REAL 2)
for i being Element of NAT st i > 2 &amp; i in dom f &amp; f is being_S-Seq holds
f | i is being_S-Seq
proofend;

theorem :: TOPREAL3:34
for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p `1 <> q `1 &amp; p `2 <> q `2 &amp; f = <*p,|[(p `1),(q `2)]|,q*> holds
( f /. 1 = p &amp; f /. (len f) = q &amp; f is being_S-Seq )
proofend;

theorem :: TOPREAL3:35
for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p `1 <> q `1 &amp; p `2 <> q `2 &amp; f = <*p,|[(q `1),(p `2)]|,q*> holds
( f /. 1 = p &amp; f /. (len f) = q &amp; f is being_S-Seq )
proofend;

theorem :: TOPREAL3:36
for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p `1 = q `1 &amp; p `2 <> q `2 &amp; f = <*p,|[(p `1),(((p `2) + (q `2)) / 2)]|,q*> holds
( f /. 1 = p &amp; f /. (len f) = q &amp; f is being_S-Seq )
proofend;

theorem :: TOPREAL3:37
for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st p `1 <> q `1 &amp; p `2 = q `2 &amp; f = <*p,|[(((p `1) + (q `1)) / 2),(p `2)]|,q*> holds
( f /. 1 = p &amp; f /. (len f) = q &amp; f is being_S-Seq )
proofend;

theorem :: TOPREAL3:38
for f being FinSequence of (TOP-REAL 2)
for i being Element of NAT st i in dom f &amp; i + 1 in dom f holds
L~ (f | (i + 1)) = (L~ (f | i)) \/ (LSeg ((f /. i),(f /. (i + 1))))
proofend;

theorem :: TOPREAL3:39
for p being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st len f >= 2 &amp; not p in L~ f holds
for n being Element of NAT st 1 <= n &amp; n <= len f holds
f /. n <> p
proofend;

theorem :: TOPREAL3:40
for q, p being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) st q <> p &amp; (LSeg (q,p)) /\ (L~ f) = {q} holds
not p in L~ f
proofend;

theorem :: TOPREAL3:41
for f being FinSequence of (TOP-REAL 2)
for m being Element of NAT st f is being_S-Seq &amp; f /. (len f) in LSeg (f,m) &amp; 1 <= m &amp; m + 1 <= len f holds
m + 1 = len f
proofend;

theorem :: TOPREAL3:42
for p1, q, p being Point of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2) st not p1 in Ball (u,r) &amp; q in Ball (u,r) &amp; p in Ball (u,r) &amp; not p in LSeg (p1,q) &amp; ( ( q `1 = p `1 &amp; q `2 <> p `2 ) or ( q `1 <> p `1 &amp; q `2 = p `2 ) ) &amp; ( p1 `1 = q `1 or p1 `2 = q `2 ) holds
(LSeg (p1,q)) /\ (LSeg (q,p)) = {q}
proofend;

theorem :: TOPREAL3:43
for p1, p, q being Point of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2) st not p1 in Ball (u,r) &amp; p in Ball (u,r) &amp; |[(p `1),(q `2)]| in Ball (u,r) &amp; not |[(p `1),(q `2)]| in LSeg (p1,p) &amp; p1 `1 = p `1 &amp; p `1 <> q `1 &amp; p `2 <> q `2 holds
((LSeg (p,|[(p `1),(q `2)]|)) \/ (LSeg (|[(p `1),(q `2)]|,q))) /\ (LSeg (p1,p)) = {p}
proofend;

theorem :: TOPREAL3:44
for p1, p, q being Point of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2) st not p1 in Ball (u,r) &amp; p in Ball (u,r) &amp; |[(q `1),(p `2)]| in Ball (u,r) &amp; not |[(q `1),(p `2)]| in LSeg (p1,p) &amp; p1 `2 = p `2 &amp; p `1 <> q `1 &amp; p `2 <> q `2 holds
((LSeg (p,|[(q `1),(p `2)]|)) \/ (LSeg (|[(q `1),(p `2)]|,q))) /\ (LSeg (p1,p)) = {p}
proofend;

theorem Th45: :: TOPREAL3:45
for n being Element of NAT
for f being FinSequence of REAL st len f = n holds
f in the carrier of (Euclid n)
proofend;

theorem :: TOPREAL3:46
for n being Element of NAT
for f being FinSequence of REAL st len f = n holds
f in the carrier of (TOP-REAL n)
proofend;