:: RCOMP_1 semantic presentation

scheme :: RCOMP_1:sch 39
s39{ P1[ set , set ] } :
ex s1 being Function of NAT , REAL st
for n being Element of NAT holds P1[n,s1 . n]
provided
E41: for n being Element of NAT ex r being real number st P1[n,r]
proof end;

theorem Th1: :: RCOMP_1:1
for X, Y being Subset of REAL st ( for r being real number st r in X holds
r in Y ) holds
X c= Y
proof end;

theorem Th2: :: RCOMP_1:2
canceled;

theorem Th3: :: RCOMP_1:3
for Y1, Y being Subset of REAL st Y1 c= Y & Y is bounded_below holds
Y1 is bounded_below
proof end;

theorem Th4: :: RCOMP_1:4
for Y1, Y being Subset of REAL st Y1 c= Y & Y is bounded_above holds
Y1 is bounded_above
proof end;

theorem Th5: :: RCOMP_1:5
for Y1, Y being Subset of REAL st Y1 c= Y & Y is bounded holds
Y1 is bounded
proof end;

definition
let g be real number , s be real number ;
func [.c1,c2.] -> Subset of REAL equals :: RCOMP_1:def 1
{ r where r is Real : ( g <= r & r <= s ) } ;
coherence
{ r where r is Real : ( g <= r & r <= s ) } is Subset of REAL
proof end;
end;

:: deftheorem Def1 defines [. RCOMP_1:def 1 :
for g, s being real number holds [.g,s.] = { r where r is Real : ( g <= r & r <= s ) } ;

definition
let g be real number , s be real number ;
func ].c1,c2.[ -> Subset of REAL equals :: RCOMP_1:def 2
{ r where r is Real : ( g < r & r < s ) } ;
coherence
{ r where r is Real : ( g < r & r < s ) } is Subset of REAL
proof end;
end;

:: deftheorem Def2 defines ]. RCOMP_1:def 2 :
for g, s being real number holds ].g,s.[ = { r where r is Real : ( g < r & r < s ) } ;

theorem Th6: :: RCOMP_1:6
canceled;

theorem Th7: :: RCOMP_1:7
canceled;

theorem Th8: :: RCOMP_1:8
for r, p, g being real number holds
( r in ].(p - g),(p + g).[ iff abs (r - p) < g )
proof end;

theorem Th9: :: RCOMP_1:9
for r, p, g being real number holds
( r in [.p,g.] iff abs ((p + g) - (2 * r)) <= g - p )
proof end;

theorem Th10: :: RCOMP_1:10
for r, p, g being real number holds
( r in ].p,g.[ iff abs ((p + g) - (2 * r)) < g - p )
proof end;

theorem Th11: :: RCOMP_1:11
for g, s being real number st g <= s holds
[.g,s.] = ].g,s.[ \/ {g,s}
proof end;

theorem Th12: :: RCOMP_1:12
for p, g being real number st p <= g holds
].g,p.[ = {}
proof end;

theorem Th13: :: RCOMP_1:13
for p, g being real number st p < g holds
[.g,p.] = {}
proof end;

theorem Th14: :: RCOMP_1:14
for p being real number holds [.p,p.] = {p}
proof end;

theorem Th15: :: RCOMP_1:15
for p, g being real number holds
( ( p < g implies ].p,g.[ <> {} ) & ( p <= g implies ( p in [.p,g.] & g in [.p,g.] ) ) & ].p,g.[ c= [.p,g.] )
proof end;

theorem Th16: :: RCOMP_1:16
for r, p, g, s being real number st r in [.p,g.] & s in [.p,g.] holds
[.r,s.] c= [.p,g.]
proof end;

theorem Th17: :: RCOMP_1:17
for r, p, g, s being real number st r in ].p,g.[ & s in ].p,g.[ holds
[.r,s.] c= ].p,g.[
proof end;

theorem Th18: :: RCOMP_1:18
for p, g being real number st p <= g holds
[.p,g.] = [.p,g.] \/ [.g,p.]
proof end;

definition
let X be Subset of REAL ;
attr a1 is compact means :Def3: :: RCOMP_1:def 3
for s1 being Real_Sequence st rng s1 c= X holds
ex s2 being Real_Sequence st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X );
end;

:: deftheorem Def3 defines compact RCOMP_1:def 3 :
for X being Subset of REAL holds
( X is compact iff for s1 being Real_Sequence st rng s1 c= X holds
ex s2 being Real_Sequence st
( s2 is subsequence of s1 & s2 is convergent & lim s2 in X ) );

definition
let X be Subset of REAL ;
attr a1 is closed means :Def4: :: RCOMP_1:def 4
for s1 being Real_Sequence st rng s1 c= X & s1 is convergent holds
lim s1 in X;
end;

:: deftheorem Def4 defines closed RCOMP_1:def 4 :
for X being Subset of REAL holds
( X is closed iff for s1 being Real_Sequence st rng s1 c= X & s1 is convergent holds
lim s1 in X );

definition
let X be Subset of REAL ;
attr a1 is open means :Def5: :: RCOMP_1:def 5
X ` is closed;
end;

:: deftheorem Def5 defines open RCOMP_1:def 5 :
for X being Subset of REAL holds
( X is open iff X ` is closed );

theorem Th19: :: RCOMP_1:19
canceled;

theorem Th20: :: RCOMP_1:20
canceled;

theorem Th21: :: RCOMP_1:21
canceled;

theorem Th22: :: RCOMP_1:22
for s, g being real number
for s1 being Real_Sequence st rng s1 c= [.s,g.] holds
s1 is bounded
proof end;

theorem Th23: :: RCOMP_1:23
for s, g being real number holds [.s,g.] is closed
proof end;

theorem Th24: :: RCOMP_1:24
for s, g being real number holds [.s,g.] is compact
proof end;

theorem Th25: :: RCOMP_1:25
for p, q being real number holds ].p,q.[ is open
proof end;

registration
let p be real number , q be real number ;
cluster ].a1,a2.[ -> open ;
coherence
].p,q.[ is open
by ;
cluster [.a1,a2.] -> closed ;
coherence
[.p,q.] is closed
by ;
end;

theorem Th26: :: RCOMP_1:26
for X being Subset of REAL st X is compact holds
X is closed
proof end;

theorem Th27: :: RCOMP_1:27
for s1 being Real_Sequence
for X being Subset of REAL st ( for p being real number st p in X holds
ex r being real number ex n being Element of NAT st
( 0 < r & ( for m being Element of NAT st n < m holds
r < abs ((s1 . m) - p) ) ) ) holds
for s2 being Real_Sequence st s2 is subsequence of s1 & s2 is convergent holds
not lim s2 in X
proof end;

theorem Th28: :: RCOMP_1:28
for X being Subset of REAL st X is compact holds
X is bounded
proof end;

theorem Th29: :: RCOMP_1:29
for X being Subset of REAL st X is bounded & X is closed holds
X is compact
proof end;

theorem Th30: :: RCOMP_1:30
for X being Subset of REAL st X <> {} & X is closed & X is bounded_above holds
upper_bound X in X
proof end;

theorem Th31: :: RCOMP_1:31
for X being Subset of REAL st X <> {} & X is closed & X is bounded_below holds
lower_bound X in X
proof end;

theorem Th32: :: RCOMP_1:32
for X being Subset of REAL st X <> {} & X is compact holds
( upper_bound X in X & lower_bound X in X )
proof end;

theorem Th33: :: RCOMP_1:33
for X being Subset of REAL st X is compact & ( for g1, g2 being real number st g1 in X & g2 in X holds
[.g1,g2.] c= X ) holds
ex p, g being real number st X = [.p,g.]
proof end;

registration
cluster open Element of K18(REAL );
existence
ex b1 being Subset of REAL st b1 is open
proof end;
end;

definition
let r be real number ;
canceled;
mode Neighbourhood of c1 -> Subset of REAL means :Def7: :: RCOMP_1:def 7
ex g being real number st
( 0 < g & it = ].(r - g),(r + g).[ );
existence
ex b1 being Subset of REAL ex g being real number st
( 0 < g & b1 = ].(r - g),(r + g).[ )
proof end;
end;

:: deftheorem Def6 RCOMP_1:def 6 :
canceled;

:: deftheorem Def7 defines Neighbourhood RCOMP_1:def 7 :
for r being real number
for b2 being Subset of REAL holds
( b2 is Neighbourhood of r iff ex g being real number st
( 0 < g & b2 = ].(r - g),(r + g).[ ) );

registration
let r be real number ;
cluster -> open Neighbourhood of a1;
coherence
for b1 being Neighbourhood of r holds b1 is open
proof end;
end;

theorem Th34: :: RCOMP_1:34
canceled;

theorem Th35: :: RCOMP_1:35
canceled;

theorem Th36: :: RCOMP_1:36
canceled;

theorem Th37: :: RCOMP_1:37
for r being real number
for N being Neighbourhood of r holds r in N
proof end;

theorem Th38: :: RCOMP_1:38
for r being real number
for N1, N2 being Neighbourhood of r ex N being Neighbourhood of r st
( N c= N1 & N c= N2 )
proof end;

theorem Th39: :: RCOMP_1:39
for X being open Subset of REAL
for r being real number st r in X holds
ex N being Neighbourhood of r st N c= X
proof end;

theorem Th40: :: RCOMP_1:40
for X being open Subset of REAL
for r being real number st r in X holds
ex g being real number st
( 0 < g & ].(r - g),(r + g).[ c= X )
proof end;

theorem Th41: :: RCOMP_1:41
for X being Subset of REAL st ( for r being real number st r in X holds
ex N being Neighbourhood of r st N c= X ) holds
X is open
proof end;

theorem Th42: :: RCOMP_1:42
for X being Subset of REAL holds
( ( for r being real number st r in X holds
ex N being Neighbourhood of r st N c= X ) iff X is open ) by , ;

theorem Th43: :: RCOMP_1:43
for X being Subset of REAL st X is open & X is bounded_above holds
not upper_bound X in X
proof end;

theorem Th44: :: RCOMP_1:44
for X being Subset of REAL st X is open & X is bounded_below holds
not lower_bound X in X
proof end;

theorem Th45: :: RCOMP_1:45
for X being Subset of REAL st X is open & X is bounded & ( for g1, g2 being real number st g1 in X & g2 in X holds
[.g1,g2.] c= X ) holds
ex p, g being real number st X = ].p,g.[
proof end;

theorem Th46: :: RCOMP_1:46
for r, s being real number holds ].r,s.[ misses {r,s}
proof end;

theorem Th47: :: RCOMP_1:47
for a, b, c being real number holds
( c in ].a,b.[ iff ( a < c & c < b ) )
proof end;

theorem Th48: :: RCOMP_1:48
for a, b, c being real number holds
( c in [.a,b.] iff ( a <= c & c <= b ) )
proof end;

theorem Th49: :: RCOMP_1:49
for a, c, d, b being real number st a <= c & d <= b holds
[.c,d.] c= [.a,b.]
proof end;

theorem Th50: :: RCOMP_1:50
for a, c, b, d being real number st a <= c & b <= d & c <= b holds
[.a,b.] \/ [.c,d.] = [.a,d.]
proof end;

theorem Th51: :: RCOMP_1:51
for a, c, b, d being real number st a <= c & b <= d & c <= b holds
[.a,b.] /\ [.c,d.] = [.c,b.]
proof end;

theorem Th52: :: RCOMP_1:52
for x, y, z, u being real number st x <= y & z <= u holds
].y,z.[ c= ].x,u.[
proof end;