:: RCOMP_2 semantic presentation

theorem :: RCOMP_2:1
canceled;

theorem :: RCOMP_2:2
for y, x, z being real number holds
( ( y < x & z < x ) iff max y,z < x ) by XXREAL_0:29, XXREAL_0:31;

definition
let g be real number ;
let s be ext-real number ;
:: original: [.
redefine func [.g,s.[ -> Subset of REAL equals :: RCOMP_2:def 1
{ r where r is Real : ( g <= r & r < s ) } ;
coherence
[.g,s.[ is Subset of REAL
proof end;
compatibility
for b1 being Subset of REAL holds
( b1 = [.g,s.[ iff b1 = { r where r is Real : ( g <= r & r < s ) } )
proof end;
end;

:: deftheorem defines [. RCOMP_2:def 1 :
for g being real number
for s being ext-real number holds [.g,s.[ = { r where r is Real : ( g <= r & r < s ) } ;

definition
let g be ext-real number ;
let s be real number ;
:: original: ].
redefine func ].g,s.] -> Subset of REAL equals :: RCOMP_2:def 2
{ r where r is Real : ( g < r & r <= s ) } ;
coherence
].g,s.] is Subset of REAL
proof end;
compatibility
for b1 being Subset of REAL holds
( b1 = ].g,s.] iff b1 = { r where r is Real : ( g < r & r <= s ) } )
proof end;
end;

:: deftheorem defines ]. RCOMP_2:def 2 :
for g being ext-real number
for s being real number holds ].g,s.] = { r where r is Real : ( g < r & r <= s ) } ;

theorem :: RCOMP_2:3
for r, p, q being real number holds
( r in [.p,q.[ iff ( p <= r & r < q ) ) by XXREAL_1:3;

theorem :: RCOMP_2:4
for r, p, q being real number holds
( r in ].p,q.] iff ( p < r & r <= q ) ) by XXREAL_1:2;

theorem :: RCOMP_2:5
for g, s being real number st g < s holds
[.g,s.[ = ].g,s.[ \/ {g} by XXREAL_1:131;

theorem :: RCOMP_2:6
for g, s being real number st g < s holds
].g,s.] = ].g,s.[ \/ {s} by XXREAL_1:132;

theorem :: RCOMP_2:7
for g being real number holds [.g,g.[ = {} by XXREAL_1:18;

theorem :: RCOMP_2:8
for g being real number holds ].g,g.] = {} by XXREAL_1:19;

theorem :: RCOMP_2:9
for p, g being real number st p <= g holds
[.g,p.[ = {} by XXREAL_1:27;

theorem :: RCOMP_2:10
for p, g being real number st p <= g holds
].g,p.] = {} by XXREAL_1:26;

theorem :: RCOMP_2:11
for g, p, h being real number st g <= p & p <= h holds
[.g,p.[ \/ [.p,h.[ = [.g,h.[ by XXREAL_1:168;

theorem :: RCOMP_2:12
for g, p, h being real number st g <= p & p <= h holds
].g,p.] \/ ].p,h.] = ].g,h.] by XXREAL_1:170;

theorem :: RCOMP_2:13
for g, p1, p2, h being real number st g <= p1 & g <= p2 & p1 <= h & p2 <= h holds
[.g,h.] = ([.g,p1.[ \/ [.p1,p2.]) \/ ].p2,h.] by XXREAL_1:179;

theorem :: RCOMP_2:14
for g, p1, p2, h being real number st g < p1 & g < p2 & p1 < h & p2 < h holds
].g,h.[ = (].g,p1.] \/ ].p1,p2.[) \/ [.p2,h.[ by XXREAL_1:180;

theorem :: RCOMP_2:15
for q1, q2, p1, p2 being real number holds [.q1,q2.[ /\ [.p1,p2.[ = [.(max q1,p1),(min q2,p2).[ by XXREAL_1:139;

theorem :: RCOMP_2:16
for q1, q2, p1, p2 being real number holds ].q1,q2.] /\ ].p1,p2.] = ].(max q1,p1),(min q2,p2).] by XXREAL_1:141;

theorem :: RCOMP_2:17
for p, q being real number holds
( ].p,q.[ c= [.p,q.[ & ].p,q.[ c= ].p,q.] & [.p,q.[ c= [.p,q.] & ].p,q.] c= [.p,q.] ) by XXREAL_1:21, XXREAL_1:22, XXREAL_1:23, XXREAL_1:24;

theorem :: RCOMP_2:18
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 :: RCOMP_2:19
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 :: RCOMP_2:20
for p, q, r being real number st p <= q & q <= r holds
[.p,q.] \/ ].q,r.] = [.p,r.] by XXREAL_1:167;

theorem :: RCOMP_2:21
for p, q, r being real number st p <= q & q <= r holds
[.p,q.[ \/ [.q,r.] = [.p,r.] by XXREAL_1:166;

theorem :: RCOMP_2:22
for q1, q2, p1, p2 being real number st [.q1,q2.[ meets [.p1,p2.[ holds
q2 >= p1 by XXREAL_1:96;

theorem :: RCOMP_2:23
for q1, q2, p1, p2 being real number st ].q1,q2.] meets ].p1,p2.] holds
q2 >= p1 by XXREAL_1:92;

theorem :: RCOMP_2:24
for q1, q2, p1, p2 being real number st [.q1,q2.[ meets [.p1,p2.[ holds
[.q1,q2.[ \/ [.p1,p2.[ = [.(min q1,p1),(max q2,p2).[ by XXREAL_1:162;

theorem :: RCOMP_2:25
for q1, q2, p1, p2 being real number st ].q1,q2.] meets ].p1,p2.] holds
].q1,q2.] \/ ].p1,p2.] = ].(min q1,p1),(max q2,p2).] by XXREAL_1:164;

theorem :: RCOMP_2:26
for p1, p2, q1, q2 being real number st [.p1,p2.[ meets [.q1,q2.[ holds
[.p1,p2.[ \ [.q1,q2.[ = [.p1,q1.[ \/ [.q2,p2.[ by XXREAL_1:198;

theorem :: RCOMP_2:27
for p1, p2, q1, q2 being real number st ].p1,p2.] meets ].q1,q2.] holds
].p1,p2.] \ ].q1,q2.] = ].p1,q1.] \/ ].q2,p2.] by XXREAL_1:199;