:: TOPREAL3 semantic presentation

theorem :: TOPREAL3:1

canceled;

theorem :: TOPREAL3:2

canceled;

theorem Th3: :: TOPREAL3:3

for r, s being real number st r < s holds
( r < (r + s) / 2 & (r + s) / 2 < s )

proof end;

Lm1: for n being Nat holds the carrier of (Euclid n) = REAL n

proof end;

theorem :: TOPREAL3:4

canceled;

theorem :: TOPREAL3:5

canceled;

theorem Th6: :: TOPREAL3:6

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

proof end;

theorem Th7: :: TOPREAL3:7

for p1, p2 being Point of (TOP-REAL 2) holds
( (p1 + p2) `1 = (p1 `1 ) + (p2 `1 ) & (p1 + p2) `2 = (p1 `2 ) + (p2 `2 ) )

proof end;

theorem :: TOPREAL3:8

for p1, p2 being Point of (TOP-REAL 2) holds
( (p1 - p2) `1 = (p1 `1 ) - (p2 `1 ) & (p1 - p2) `2 = (p1 `2 ) - (p2 `2 ) )

proof end;

theorem Th9: :: TOPREAL3:9

for p being Point of (TOP-REAL 2)
for r being real number holds
( (r * p) `1 = r * (p `1 ) & (r * p) `2 = r * (p `2 ) )

proof end;

theorem Th10: :: TOPREAL3:10

for p1, p2 being Point of (TOP-REAL 2)
for r1, s1, r2, s2 being real number st p1 = <*r1,s1*> & p2 = <*r2,s2*> holds
( p1 + p2 = <*(r1 + r2),(s1 + s2)*> & p1 - p2 = <*(r1 - r2),(s1 - s2)*> )

proof end;

theorem Th11: :: TOPREAL3:11

for p, q being Point of (TOP-REAL 2) st p `1 = q `1 & p `2 = q `2 holds
p = q

proof end;

theorem Th12: :: TOPREAL3:12

for p1, p2 being Point of (TOP-REAL 2)
for u1, u2 being Point of (Euclid 2) st u1 = p1 & u2 = p2 holds
(Pitag_dist 2) . u1,u2 = sqrt ((((p1 `1 ) - (p2 `1 )) ^2 ) + (((p1 `2 ) - (p2 `2 )) ^2 ))

proof end;

theorem Th13: :: TOPREAL3:13

for n being Nat holds the carrier of (TOP-REAL n) = the carrier of (Euclid n)

proof end;

theorem :: TOPREAL3:14

canceled;

theorem Th15: :: TOPREAL3:15

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

proof end;

theorem Th16: :: TOPREAL3:16

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

proof end;

theorem :: TOPREAL3:17

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

proof end;

theorem :: TOPREAL3:18

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

proof end;

theorem :: TOPREAL3:19

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

proof end;

theorem :: TOPREAL3:20

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

proof end;

theorem Th21: :: TOPREAL3:21

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*> & i <> 0 & j > i + 1 holds
LSeg f,j = {}

proof end;

theorem :: TOPREAL3:22

canceled;

theorem :: TOPREAL3:23

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)

proof end;

theorem Th24: :: TOPREAL3:24

for f being FinSequence of (TOP-REAL 2)
for i, j being Nat st i in dom f & j in dom (f | i) & j + 1 in dom (f | i) holds
LSeg f,j = LSeg (f | i),j

proof end;

theorem :: TOPREAL3:25

for f, h being FinSequence of (TOP-REAL 2)
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 Th26: :: TOPREAL3:26

for n being Nat
for f being FinSequence of (TOP-REAL n)
for i being Nat holds LSeg f,i c= L~ f

proof end;

theorem :: TOPREAL3:27

for f being FinSequence of (TOP-REAL 2)
for i being Nat holds L~ (f | i) c= L~ f

proof end;

theorem Th28: :: TOPREAL3:28

for r being real number
for n being Nat
for p1, p2 being Point of (TOP-REAL n)
for u being Point of (Euclid n) st p1 in Ball u,r & p2 in Ball u,r holds
LSeg p1,p2 c= Ball u,r

proof end;

theorem :: TOPREAL3:29

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 & p1 = |[r1,s1]| & p2 = |[r2,s2]| & p = |[r2,s1]| & p2 in Ball u,r holds
p in Ball u,r

proof end;

theorem :: TOPREAL3:30

for s, r1, r, s1 being real number
for u being Point of (Euclid 2) 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 :: TOPREAL3:31

for r1, s, r, s1 being real number
for u being Point of (Euclid 2) 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 :: TOPREAL3:32

for r1, s1, s2, r2, r being real number
for u being Point of (Euclid 2) st r1 <> s1 & s2 <> r2 & |[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 :: TOPREAL3:33

for f being FinSequence of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2)
for m being Nat st not f /. 1 in Ball u,r & 1 <= m & m <= (len f) - 1 & LSeg f,m meets Ball u,r & ( 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 :: TOPREAL3:34

for q, p2, p being Point of (TOP-REAL 2) 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 :: TOPREAL3:35

for q, p2, p being Point of (TOP-REAL 2) 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 Th36: :: TOPREAL3:36

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 )]|}

proof end;

theorem Th37: :: TOPREAL3:37

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 )]|}

proof end;

theorem Th38: :: TOPREAL3:38

for p, q being Point of (TOP-REAL 2) 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 Th39: :: TOPREAL3:39

for p, q being Point of (TOP-REAL 2) 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 :: TOPREAL3:40

for f being FinSequence of (TOP-REAL 2)
for i being Nat st i > 2 & i in dom f & f is_S-Seq holds
f | i is_S-Seq

proof end;

theorem :: TOPREAL3:41

for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) 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_S-Seq )

proof end;

theorem :: TOPREAL3:42

for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) 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_S-Seq )

proof end;

theorem :: TOPREAL3:43

for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) 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_S-Seq )

proof end;

theorem :: TOPREAL3:44

for p, q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) 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_S-Seq )

proof end;

theorem :: TOPREAL3:45

for f being FinSequence of (TOP-REAL 2)
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 :: TOPREAL3:46

for p being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2) 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 :: TOPREAL3:47

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

proof end;

theorem :: TOPREAL3:48

for q being Point of (TOP-REAL 2)
for f being FinSequence of (TOP-REAL 2)
for r being real number
for u being Point of (Euclid 2)
for m being Nat st f is_S-Seq & not f /. 1 in Ball u,r & q in Ball u,r & f /. (len f) in LSeg f,m & 1 <= m & m + 1 <= len f & LSeg f,m meets Ball u,r holds
m + 1 = len f

proof end;

theorem :: TOPREAL3:49

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 & 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 :: TOPREAL3:50

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 & p in Ball u,r & |[(p `1 ),(q `2 )]| in Ball u,r & q 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 :: TOPREAL3:51

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 & p in Ball u,r & |[(q `1 ),(p `2 )]| in Ball u,r & q 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;