:: by Yatsuka Nakamura and Jaros{\l}aw Kotowicz

::

:: Received August 24, 1992

:: Copyright (c) 1992-2021 Association of Mizar Users

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 (TOP-REAL 2) holds

( (p1 + p2) `1 = (p1 `1) + (p2 `1) & (p1 + p2) `2 = (p1 `2) + (p2 `2) )

( (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 (TOP-REAL 2) holds

( (p1 - p2) `1 = (p1 `1) - (p2 `1) & (p1 - p2) `2 = (p1 `2) - (p2 `2) )

( (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 (TOP-REAL 2)

for r being Real holds

( (r * p) `1 = r * (p `1) & (r * p) `2 = r * (p `2) )

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 (TOP-REAL 2)

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)*> )

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 Th7: :: TOPREAL3:7

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))

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 Th9: :: TOPREAL3:9

for r, r1, s1 being Real 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]|)

{ 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 Th10: :: TOPREAL3:10

for r, r1, s1 being Real 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]|)

{ 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 :: TOPREAL3:11

for p being Point of (TOP-REAL 2)

for r, r1, s1 being Real st p in LSeg (|[r,r1]|,|[r,s1]|) holds

p `1 = r

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 (TOP-REAL 2)

for r, r1, s1 being Real st p in LSeg (|[r1,r]|,|[s1,r]|) holds

p `2 = r

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 (TOP-REAL 2) st p `1 <> q `1 & p `2 = q `2 holds

|[(((p `1) + (q `1)) / 2),(p `2)]| in LSeg (p,q)

|[(((p `1) + (q `1)) / 2),(p `2)]| in LSeg (p,q)

proof end;

theorem :: TOPREAL3:14

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)

|[(p `1),(((p `2) + (q `2)) / 2)]| in LSeg (p,q)

proof end;

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*> & i <> 0 & j > i + 1 holds

LSeg (f,j) = {}

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 :: TOPREAL3:16

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))

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 Th17: :: TOPREAL3:17

for f being FinSequence of (TOP-REAL 2)

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)

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 (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)

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 Th21: :: TOPREAL3:21

for r being Real

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)

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 :: TOPREAL3:22

for p, p1, p2 being Point of (TOP-REAL 2)

for r, r1, r2, s1, s2 being Real

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)

for r, r1, r2, s1, s2 being Real

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 :: TOPREAL3:23

for r, r1, s, s1 being Real

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)

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 :: TOPREAL3:24

for r, r1, s, s1 being Real

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)

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 :: TOPREAL3:25

for r, r1, r2, s1, s2 being Real

for u being Point of (Euclid 2) 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)

for u being Point of (Euclid 2) 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 (TOP-REAL 2)

for r being Real

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 & ( 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)

for r being Real

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 & ( 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 (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}

((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 (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}

((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 (TOP-REAL 2) 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 (TOP-REAL 2) 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 (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)]|}

(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 (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)]|}

(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 (TOP-REAL 2)

for i being Nat st i > 2 & i in dom f & f is being_S-Seq holds

f | i is being_S-Seq

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 (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 being_S-Seq )

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 being_S-Seq )

proof end;

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 & p `2 <> q `2 & f = <*p,|[(q `1),(p `2)]|,q*> holds

( f /. 1 = p & f /. (len f) = q & f is being_S-Seq )

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 being_S-Seq )

proof end;

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 & 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 )

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 being_S-Seq )

proof end;

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 & 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 )

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 being_S-Seq )

proof end;

theorem :: TOPREAL3:38

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))))

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 (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

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 :: TOPREAL3:40

for p, q 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

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 :: TOPREAL3:41

for f being FinSequence of (TOP-REAL 2)

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

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 (TOP-REAL 2)

for r being Real

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}

for r being Real

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 :: TOPREAL3:43

for p, p1, q being Point of (TOP-REAL 2)

for r being Real

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) & 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}

for r being Real

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) & 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 (TOP-REAL 2)

for r being Real

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) & 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}

for r being Real

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) & 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;