:: Basic Properties of Connecting Points with Line Segments in ${\calE}^2_{\rm T}$
:: by Yatsuka Nakamura and Jaros{\l}aw Kotowicz
::
:: Copyright (c) 1992-2021 Association of Mizar Users

theorem :: TOPREAL3:1
canceled;

Th1: for x, y, z being object holds
( 1 in dom <*x,y,z*> & 2 in dom <*x,y,z*> & 3 in dom <*x,y,z*> )

by FINSEQ_1:81;

theorem Th2: :: TOPREAL3:2
for p1, p2 being Point of () holds
( (p1 + p2) 1 = (p1 1) + (p2 1) & (p1 + p2) 2 = (p1 2) + (p2 2) )
proof end;

theorem :: TOPREAL3:3
for p1, p2 being Point of () holds
( (p1 - p2) 1 = (p1 1) - (p2 1) & (p1 - p2) 2 = (p1 2) - (p2 2) )
proof end;

theorem Th4: :: TOPREAL3:4
for p being Point of ()
for r being Real holds
( (r * p) 1 = r * (p 1) & (r * p) 2 = r * (p 2) )
proof end;

theorem Th5: :: TOPREAL3:5
for p1, p2 being Point of ()
for r1, r2, s1, s2 being Real st p1 = <*r1,s1*> & p2 = <*r2,s2*> holds
( p1 + p2 = <*(r1 + r2),(s1 + s2)*> & p1 - p2 = <*(r1 - r2),(s1 - s2)*> )
proof end;

theorem Th6: :: TOPREAL3:6
for p, q being Point of () st p 1 = q 1 & p 2 = q 2 holds
p = q
proof end;

theorem Th7: :: TOPREAL3:7
for p1, p2 being Point of ()
for u1, u2 being Point of () st u1 = p1 & u2 = p2 holds
() . (u1,u2) = sqrt ((((p1 1) - (p2 1)) ^2) + (((p1 2) - (p2 2)) ^2))
proof end;

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

theorem Th9: :: TOPREAL3:9
for r, r1, s1 being Real st r1 <= s1 holds
{ p1 where p1 is Point of () : ( p1 1 = r & r1 <= p1 2 & p1 2 <= s1 ) } = LSeg (|[r,r1]|,|[r,s1]|)
proof end;

theorem Th10: :: TOPREAL3:10
for r, r1, s1 being Real st r1 <= s1 holds
{ p1 where p1 is Point of () : ( p1 2 = r & r1 <= p1 1 & p1 1 <= s1 ) } = LSeg (|[r1,r]|,|[s1,r]|)
proof end;

theorem :: TOPREAL3:11
for p being Point of ()
for r, r1, s1 being Real st p in LSeg (|[r,r1]|,|[r,s1]|) holds
p 1 = r
proof end;

theorem :: TOPREAL3:12
for p being Point of ()
for r, r1, s1 being Real st p in LSeg (|[r1,r]|,|[s1,r]|) holds
p 2 = r
proof end;

theorem :: TOPREAL3:13
for p, q being Point of () st p 1 <> q 1 & p 2 = q 2 holds
|[(((p 1) + (q 1)) / 2),(p 2)]| in LSeg (p,q)
proof end;

theorem :: TOPREAL3:14
for p, q being Point of () st p 1 = q 1 & p 2 <> q 2 holds
|[(p 1),(((p 2) + (q 2)) / 2)]| in LSeg (p,q)
proof end;

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

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

theorem Th17: :: TOPREAL3:17
for f being FinSequence of ()
for i, j being Nat st j in dom (f | i) & j + 1 in dom (f | i) holds
LSeg (f,j) = LSeg ((f | i),j)
proof end;

theorem :: TOPREAL3:18
for f, h being FinSequence of ()
for j being Nat st j in dom f & j + 1 in dom f holds
LSeg ((f ^ h),j) = LSeg (f,j)
proof end;

theorem Th19: :: TOPREAL3:19
for n being Nat
for f being FinSequence of ()
for i being Nat holds LSeg (f,i) c= L~ f
proof end;

theorem :: TOPREAL3:20
for f being FinSequence of ()
for i being Nat holds L~ (f | i) c= L~ f
proof end;

theorem Th21: :: TOPREAL3:21
for r being Real
for N being Nat
for p1, p2 being Point of ()
for u being Point of () st p1 in Ball (u,r) & p2 in Ball (u,r) holds
LSeg (p1,p2) c= Ball (u,r)
proof end;

theorem :: TOPREAL3:22
for p, p1, p2 being Point of ()
for r, r1, r2, s1, s2 being Real
for u being Point of () st u = p1 & p1 = |[r1,s1]| & p2 = |[r2,s2]| & p = |[r2,s1]| & p2 in Ball (u,r) holds
p in Ball (u,r)
proof end;

theorem :: TOPREAL3:23
for r, r1, s, s1 being Real
for u being Point of () st |[s,r1]| in Ball (u,r) & |[s,s1]| in Ball (u,r) holds
|[s,((r1 + s1) / 2)]| in Ball (u,r)
proof end;

theorem :: TOPREAL3:24
for r, r1, s, s1 being Real
for u being Point of () st |[r1,s]| in Ball (u,r) & |[s1,s]| in Ball (u,r) holds
|[((r1 + s1) / 2),s]| in Ball (u,r)
proof end;

theorem :: TOPREAL3:25
for r, r1, r2, s1, s2 being Real
for u being Point of () st |[r1,r2]| in Ball (u,r) & |[s1,s2]| in Ball (u,r) & not |[r1,s2]| in Ball (u,r) holds
|[s1,r2]| in Ball (u,r)
proof end;

theorem :: TOPREAL3:26
for f being FinSequence of ()
for r being Real
for u being Point of ()
for m being Nat st not f /. 1 in Ball (u,r) & 1 <= m & m <= (len f) - 1 & ( for i being Nat st 1 <= i & i <= (len f) - 1 & (LSeg (f,i)) /\ (Ball (u,r)) <> {} holds
m <= i ) holds
not f /. m in Ball (u,r)
proof end;

theorem :: TOPREAL3:27
for q, p2, p being Point of () st q 2 = p2 2 & p 2 <> p2 2 holds
((LSeg (p2,|[(p2 1),(p 2)]|)) \/ (LSeg (|[(p2 1),(p 2)]|,p))) /\ (LSeg (q,p2)) = {p2}
proof end;

theorem :: TOPREAL3:28
for q, p2, p being Point of () st q 1 = p2 1 & p 1 <> p2 1 holds
((LSeg (p2,|[(p 1),(p2 2)]|)) \/ (LSeg (|[(p 1),(p2 2)]|,p))) /\ (LSeg (q,p2)) = {p2}
proof end;

theorem Th29: :: TOPREAL3:29
for p, q being Point of () holds (LSeg (p,|[(p 1),(q 2)]|)) /\ (LSeg (|[(p 1),(q 2)]|,q)) = {|[(p 1),(q 2)]|}
proof end;

theorem Th30: :: TOPREAL3:30
for p, q being Point of () holds (LSeg (p,|[(q 1),(p 2)]|)) /\ (LSeg (|[(q 1),(p 2)]|,q)) = {|[(q 1),(p 2)]|}
proof end;

theorem Th31: :: TOPREAL3:31
for p, q being Point of () st p 1 = q 1 & 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)]|}
proof end;

theorem Th32: :: TOPREAL3:32
for p, q being Point of () st p 1 <> q 1 & 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)]|}
proof end;

theorem :: TOPREAL3:33
for f being FinSequence of ()
for i being Nat st i > 2 & i in dom f & f is being_S-Seq holds
f | i is being_S-Seq
proof end;

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

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

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

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

theorem :: TOPREAL3:38
for f being FinSequence of ()
for i being Nat st i in dom f & i + 1 in dom f holds
L~ (f | (i + 1)) = (L~ (f | i)) \/ (LSeg ((f /. i),(f /. (i + 1))))
proof end;

theorem :: TOPREAL3:39
for p being Point of ()
for f being FinSequence of () st len f >= 2 & not p in L~ f holds
for n being Nat st 1 <= n & n <= len f holds
f /. n <> p
proof end;

theorem :: TOPREAL3:40
for p, q being Point of ()
for f being FinSequence of () st q <> p & (LSeg (q,p)) /\ (L~ f) = {q} holds
not p in L~ f
proof end;

theorem :: TOPREAL3:41
for f being FinSequence of ()
for m being Nat st f is being_S-Seq & f /. (len f) in LSeg (f,m) & 1 <= m & m + 1 <= len f holds
m + 1 = len f
proof end;

theorem :: TOPREAL3:42
for p, p1, q being Point of ()
for r being Real
for u being Point of () st not p1 in Ball (u,r) & q in Ball (u,r) & p in Ball (u,r) & not p in LSeg (p1,q) & ( ( q 1 = p 1 & q 2 <> p 2 ) or ( q 1 <> p 1 & q 2 = p 2 ) ) & ( p1 1 = q 1 or p1 2 = q 2 ) holds
(LSeg (p1,q)) /\ (LSeg (q,p)) = {q}
proof end;

theorem :: TOPREAL3:43
for p, p1, q being Point of ()
for r being Real
for u being Point of () st not p1 in Ball (u,r) & p in Ball (u,r) & |[(p 1),(q 2)]| in Ball (u,r) & not |[(p 1),(q 2)]| in LSeg (p1,p) & p1 1 = p 1 & p 1 <> q 1 & p 2 <> q 2 holds
((LSeg (p,|[(p 1),(q 2)]|)) \/ (LSeg (|[(p 1),(q 2)]|,q))) /\ (LSeg (p1,p)) = {p}
proof end;

theorem :: TOPREAL3:44
for p, p1, q being Point of ()
for r being Real
for u being Point of () st not p1 in Ball (u,r) & p in Ball (u,r) & |[(q 1),(p 2)]| in Ball (u,r) & not |[(q 1),(p 2)]| in LSeg (p1,p) & p1 2 = p 2 & p 1 <> q 1 & p 2 <> q 2 holds
((LSeg (p,|[(q 1),(p 2)]|)) \/ (LSeg (|[(q 1),(p 2)]|,q))) /\ (LSeg (p1,p)) = {p}
proof end;

theorem Th45: :: TOPREAL3:45
for n being Nat
for f being FinSequence of REAL st len f = n holds
f in the carrier of ()
proof end;

theorem :: TOPREAL3:46
for n being Nat
for f being FinSequence of REAL st len f = n holds
f in the carrier of ()
proof end;