:: Projections in n-Dimensional Euclidean Space to Each Coordinates
:: by Roman Matuszewski and Yatsuka Nakamura
::
:: Received November 3, 1997
:: Copyright (c) 1997-2012 Association of Mizar Users


begin

Lm1:  for n, i being Nat
for p being Element of (TOP-REAL n) ex q being Real ex g being FinSequence of REAL st
( g = p & q = g /. i )
proofend;

registration
let n be Nat;
cluster  ->  REAL -valued for Element of the carrier of (TOP-REAL n);
coherence
 for b1 being Element of (TOP-REAL n) holds b1 is REAL -valued
proofend;
end;

theorem :: JORDAN2B:1
for i being Element of NAT
for n being Nat ex f being Function of (TOP-REAL n),R^1 st
for p being Element of (TOP-REAL n) holds f . p = p /. i
proofend;

theorem :: JORDAN2B:2
for n, i being Element of NAT st i in Seg n holds
(0. (TOP-REAL n)) /. i = 0
proofend;

theorem :: JORDAN2B:3
for n being Element of NAT
for r being real number
for p being Point of (TOP-REAL n)
for i being Element of NAT st i in Seg n holds
(r * p) /. i = r * (p /. i)
proofend;

theorem :: JORDAN2B:4
for n being Element of NAT
for p being Point of (TOP-REAL n)
for i being Element of NAT st i in Seg n holds
(- p) /. i = - (p /. i)
proofend;

theorem :: JORDAN2B:5
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for i being Element of NAT st i in Seg n holds
(p1 + p2) /. i = (p1 /. i) + (p2 /. i)
proofend;

theorem Th6: :: JORDAN2B:6
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for i being Element of NAT st i in Seg n holds
(p1 - p2) /. i = (p1 /. i) - (p2 /. i)
proofend;

theorem Th7: :: JORDAN2B:7
for i, n being Element of NAT st i <= n holds
(0* n) | i = 0* i
proofend;

theorem Th8: :: JORDAN2B:8
for n, i being Element of NAT holds (0* n) /^ i = 0* (n -' i)
proofend;

theorem Th9: :: JORDAN2B:9
for i being Element of NAT holds Sum (0* i) = 0
proofend;

theorem Th10: :: JORDAN2B:10
for i, n being Element of NAT
for r being Real st i in Seg n holds
Sum ((0* n) +* (i,r)) = r
proofend;

theorem Th11: :: JORDAN2B:11
for n being Element of NAT
for q being Element of REAL n
for p being Point of (TOP-REAL n)
for i being Element of NAT st i in Seg n &amp; q = p holds
( p /. i <= |.q.| &amp; (p /. i) ^2 <= |.q.| ^2 )
proofend;

begin

theorem Th12: :: JORDAN2B:12
for n being Element of NAT
for s1 being real number
for P being Subset of (TOP-REAL n)
for i being Element of NAT st P =  {  p where p is Point of (TOP-REAL n) : s1 > p /. i  }  &amp; i in Seg n holds
P is open
proofend;

theorem Th13: :: JORDAN2B:13
for n being Element of NAT
for s1 being real number
for P being Subset of (TOP-REAL n)
for i being Element of NAT st P =  {  p where p is Point of (TOP-REAL n) : s1 < p /. i  }  &amp; i in Seg n holds
P is open
proofend;

theorem Th14: :: JORDAN2B:14
for n being Element of NAT
for P being Subset of (TOP-REAL n)
for a, b being real number
for i being Element of NAT st P =  {  p where p is Element of (TOP-REAL n) : ( a < p /. i &amp; p /. i < b )  }  &amp; i in Seg n holds
P is open
proofend;

theorem Th15: :: JORDAN2B:15
for n being Element of NAT
for a, b being real number
for f being Function of (TOP-REAL n),R^1
for i being Element of NAT st ( for p being Element of (TOP-REAL n) holds f . p = p /. i ) holds
f "  {  s where s is Real : ( a < s &amp; s < b )  }  =  {  p where p is Element of (TOP-REAL n) : ( a < p /. i &amp; p /. i < b )  }
proofend;

theorem Th16: :: JORDAN2B:16
for X being non empty TopSpace
for M being non empty MetrSpace
for f being Function of X,(TopSpaceMetr M) st ( for r being real number
for u being Element of M
for P being Subset of (TopSpaceMetr M) st r > 0 &amp; P = Ball (u,r) holds
f " P is open ) holds
f is continuous
proofend;

theorem Th17: :: JORDAN2B:17
for u being Point of RealSpace
for r, u1 being real number st u1 = u holds
Ball (u,r) =  {  s where s is Real : ( u1 - r < s &amp; s < u1 + r )  }
proofend;

theorem Th18: :: JORDAN2B:18
for n being Element of NAT
for f being Function of (TOP-REAL n),R^1
for i being Element of NAT st i in Seg n &amp; ( for p being Element of (TOP-REAL n) holds f . p = p /. i ) holds
f is continuous
proofend;

begin

theorem :: JORDAN2B:19
for s being Real holds abs <*s*> = <*(abs s)*>
proofend;

theorem Th20: :: JORDAN2B:20
for p being Element of (TOP-REAL 1) ex r being Real st p = <*r*>
proofend;

notation
let r be real number ;
synonym |[r]| for <*r*>;
end;

definition
let r be real number ;
:: original:|[
redefine func|[r]| ->  Point of (TOP-REAL 1);
coherence
 |[r]| is Point of (TOP-REAL 1)
proofend;
end;

theorem :: JORDAN2B:21
for r being real number
for s being Real holds s * |[r]| = |[(s * r)]| by RVSUM_1:47;

theorem :: JORDAN2B:22
for r1, r2 being real number holds |[(r1 + r2)]| = |[r1]| + |[r2]| by RVSUM_1:13;

theorem :: JORDAN2B:23
for r1, r2 being real number st |[r1]| = |[r2]| holds
r1 = r2 by FINSEQ_1:76;

theorem Th24: :: JORDAN2B:24
for P being Subset of R^1
for b being real number st P =  {  s where s is Real : s < b  }  holds
P is open
proofend;

theorem Th25: :: JORDAN2B:25
for P being Subset of R^1
for a being real number st P =  {  s where s is Real : a < s  }  holds
P is open
proofend;

theorem Th26: :: JORDAN2B:26
for P being Subset of R^1
for a, b being real number st P =  {  s where s is Real : ( a < s &amp; s < b )  }  holds
P is open
proofend;

theorem Th27: :: JORDAN2B:27
for u being Point of (Euclid 1)
for r, u1 being real number st <*u1*> = u holds
Ball (u,r) =  {  <*s*> where s is Real : ( u1 - r < s &amp; s < u1 + r )  }
proofend;

theorem :: JORDAN2B:28
for f being Function of (TOP-REAL 1),R^1 st ( for p being Element of (TOP-REAL 1) holds f . p = p /. 1 ) holds
f is being_homeomorphism
proofend;

theorem Th29: :: JORDAN2B:29
for p being Element of (TOP-REAL 2) holds
( p /. 1 = p `1 &amp; p /. 2 = p `2 )
proofend;

theorem :: JORDAN2B:30
for p being Element of (TOP-REAL 2) holds
( p /. 1 = proj1 . p &amp; p /. 2 = proj2 . p )
proofend;