:: Half Open Intervals in Real Numbers
:: by Yatsuka Nakamura
::
:: Received February 1, 2002
:: Copyright (c) 2002 Association of Mizar Users

:: RCOMP_2 semantic presentation

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_2:1
canceled;

theorem :: RCOMP_2:2
canceled;

theorem :: RCOMP_2:3
canceled;

theorem :: RCOMP_2:4
canceled;

theorem :: RCOMP_2:5
canceled;

theorem :: RCOMP_2:6
canceled;

theorem :: RCOMP_2:7
canceled;

theorem :: RCOMP_2:8
canceled;

theorem :: RCOMP_2:9
canceled;

theorem :: RCOMP_2:10
canceled;

theorem :: RCOMP_2:11
canceled;

theorem :: RCOMP_2:12
canceled;

theorem :: RCOMP_2:13
canceled;

theorem :: RCOMP_2:14
canceled;

theorem :: RCOMP_2:15
canceled;

theorem :: RCOMP_2:16
canceled;

theorem :: RCOMP_2:17
canceled;

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;