:: XXREAL_1 semantic presentation
begin
scheme :: XXREAL_1:sch 1
Conti{ P1[ set ], P2[ set ] } :
ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
provided
A1: for r, s being ext-real number st P1[r] & P2[s] holds
r <= s
proof
set A = { a where a is Element of ExtREAL : P1[a] } ;
set B = { a where a is Element of ExtREAL : P2[a] } ;
reconsider X = { a where a is Element of ExtREAL : P1[a] } /\ REAL, Y = { a where a is Element of ExtREAL : P2[a] } /\ REAL as Subset of REAL by XBOOLE_1:17;
percases ( X = {} or Y = {} or ( X <> {} & Y <> {} ) ) ;
supposeA2: X = {} ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
percases ( +infty in { a where a is Element of ExtREAL : P1[a] } or not +infty in { a where a is Element of ExtREAL : P1[a] } ) ;
supposeA3: +infty in { a where a is Element of ExtREAL : P1[a] } ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
take +infty ; ::_thesis: ( ( for r being ext-real number st P1[r] holds
r <= +infty ) & ( for r being ext-real number st P2[r] holds
+infty <= r ) )
thus for r being ext-real number st P1[r] holds
r <= +infty by XXREAL_0:3; ::_thesis: for r being ext-real number st P2[r] holds
+infty <= r
ex a being Element of ExtREAL st
( a = +infty & P1[a] ) by A3;
hence for r being ext-real number st P2[r] holds
+infty <= r by A1; ::_thesis: verum
end;
supposeA4: not +infty in { a where a is Element of ExtREAL : P1[a] } ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
take -infty ; ::_thesis: ( ( for r being ext-real number st P1[r] holds
r <= -infty ) & ( for r being ext-real number st P2[r] holds
-infty <= r ) )
thus for r being ext-real number st P1[r] holds
r <= -infty ::_thesis: for r being ext-real number st P2[r] holds
-infty <= r
proof
let r be ext-real number ; ::_thesis: ( P1[r] implies r <= -infty )
assume A5: P1[r] ; ::_thesis: r <= -infty
r in ExtREAL by XXREAL_0:def_1;
then A6: r in { a where a is Element of ExtREAL : P1[a] } by A5;
percases ( r = -infty or r in REAL ) by A4, A6, XXREAL_0:14;
suppose r = -infty ; ::_thesis: r <= -infty
hence r <= -infty ; ::_thesis: verum
end;
suppose r in REAL ; ::_thesis: r <= -infty
hence r <= -infty by A2, A6, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
thus for r being ext-real number st P2[r] holds
-infty <= r by XXREAL_0:5; ::_thesis: verum
end;
end;
end;
supposeA7: Y = {} ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
percases ( -infty in { a where a is Element of ExtREAL : P2[a] } or not -infty in { a where a is Element of ExtREAL : P2[a] } ) ;
supposeA8: -infty in { a where a is Element of ExtREAL : P2[a] } ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
take -infty ; ::_thesis: ( ( for r being ext-real number st P1[r] holds
r <= -infty ) & ( for r being ext-real number st P2[r] holds
-infty <= r ) )
ex a being Element of ExtREAL st
( a = -infty & P2[a] ) by A8;
hence for r being ext-real number st P1[r] holds
r <= -infty by A1; ::_thesis: for r being ext-real number st P2[r] holds
-infty <= r
thus for r being ext-real number st P2[r] holds
-infty <= r by XXREAL_0:5; ::_thesis: verum
end;
supposeA9: not -infty in { a where a is Element of ExtREAL : P2[a] } ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
take +infty ; ::_thesis: ( ( for r being ext-real number st P1[r] holds
r <= +infty ) & ( for r being ext-real number st P2[r] holds
+infty <= r ) )
thus for r being ext-real number st P1[r] holds
r <= +infty by XXREAL_0:3; ::_thesis: for r being ext-real number st P2[r] holds
+infty <= r
let r be ext-real number ; ::_thesis: ( P2[r] implies +infty <= r )
assume A10: P2[r] ; ::_thesis: +infty <= r
r in ExtREAL by XXREAL_0:def_1;
then A11: r in { a where a is Element of ExtREAL : P2[a] } by A10;
percases ( r = +infty or r in REAL ) by A9, A11, XXREAL_0:14;
suppose r = +infty ; ::_thesis: +infty <= r
hence +infty <= r ; ::_thesis: verum
end;
suppose r in REAL ; ::_thesis: +infty <= r
hence +infty <= r by A7, A11, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
end;
end;
supposethat A12: X <> {} and
A13: Y <> {} ; ::_thesis: ex s being ext-real number st
( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
for x, y being real number st x in X & y in Y holds
x <= y
proof
let x, y be real number ; ::_thesis: ( x in X & y in Y implies x <= y )
assume x in X ; ::_thesis: ( not y in Y or x <= y )
then x in { a where a is Element of ExtREAL : P1[a] } by XBOOLE_0:def_4;
then A14: ex a being Element of ExtREAL st
( a = x & P1[a] ) ;
assume y in Y ; ::_thesis: x <= y
then y in { a where a is Element of ExtREAL : P2[a] } by XBOOLE_0:def_4;
then ex a being Element of ExtREAL st
( a = y & P2[a] ) ;
hence x <= y by A1, A14; ::_thesis: verum
end;
then consider s being real number such that
A15: for x, y being real number st x in X & y in Y holds
( x <= s & s <= y ) by AXIOMS:1;
reconsider s = s as ext-real number ;
take s ; ::_thesis: ( ( for r being ext-real number st P1[r] holds
r <= s ) & ( for r being ext-real number st P2[r] holds
s <= r ) )
thus for r being ext-real number st P1[r] holds
r <= s ::_thesis: for r being ext-real number st P2[r] holds
s <= r
proof
let r be ext-real number ; ::_thesis: ( P1[r] implies r <= s )
consider x being Element of REAL such that
A16: x in Y by A13, SUBSET_1:4;
x in { a where a is Element of ExtREAL : P2[a] } by A16, XBOOLE_0:def_4;
then A17: ex a being Element of ExtREAL st
( x = a & P2[a] ) ;
assume A18: P1[r] ; ::_thesis: r <= s
percases ( r in REAL or r = -infty ) by A1, A17, A18, XXREAL_0:13;
supposeA19: r in REAL ; ::_thesis: r <= s
then reconsider r = r as real number ;
r is Element of ExtREAL by XXREAL_0:def_1;
then r in { a where a is Element of ExtREAL : P1[a] } by A18;
then r in X by A19, XBOOLE_0:def_4;
hence r <= s by A15, A16; ::_thesis: verum
end;
suppose r = -infty ; ::_thesis: r <= s
hence r <= s by XXREAL_0:5; ::_thesis: verum
end;
end;
end;
let r be ext-real number ; ::_thesis: ( P2[r] implies s <= r )
consider x being Element of REAL such that
A20: x in X by A12, SUBSET_1:4;
x in { a where a is Element of ExtREAL : P1[a] } by A20, XBOOLE_0:def_4;
then A21: ex a being Element of ExtREAL st
( x = a & P1[a] ) ;
assume A22: P2[r] ; ::_thesis: s <= r
percases ( r in REAL or r = +infty ) by A1, A21, A22, XXREAL_0:10;
supposeA23: r in REAL ; ::_thesis: s <= r
then reconsider r = r as real number ;
r is Element of ExtREAL by XXREAL_0:def_1;
then r in { a where a is Element of ExtREAL : P2[a] } by A22;
then r in Y by A23, XBOOLE_0:def_4;
hence s <= r by A15, A20; ::_thesis: verum
end;
suppose r = +infty ; ::_thesis: s <= r
hence s <= r by XXREAL_0:3; ::_thesis: verum
end;
end;
end;
end;
end;
begin
definition
let r, s be ext-real number ;
func[.r,s.] -> set equals :: XXREAL_1:def 1
{ a where a is Element of ExtREAL : ( r <= a & a <= s ) } ;
correctness
coherence
{ a where a is Element of ExtREAL : ( r <= a & a <= s ) } is set ;
;
func[.r,s.[ -> set equals :: XXREAL_1:def 2
{ a where a is Element of ExtREAL : ( r <= a & a < s ) } ;
correctness
coherence
{ a where a is Element of ExtREAL : ( r <= a & a < s ) } is set ;
;
func].r,s.] -> set equals :: XXREAL_1:def 3
{ a where a is Element of ExtREAL : ( r < a & a <= s ) } ;
correctness
coherence
{ a where a is Element of ExtREAL : ( r < a & a <= s ) } is set ;
;
func].r,s.[ -> set equals :: XXREAL_1:def 4
{ a where a is Element of ExtREAL : ( r < a & a < s ) } ;
correctness
coherence
{ a where a is Element of ExtREAL : ( r < a & a < s ) } is set ;
;
end;
:: deftheorem defines [. XXREAL_1:def_1_:_
for r, s being ext-real number holds [.r,s.] = { a where a is Element of ExtREAL : ( r <= a & a <= s ) } ;
:: deftheorem defines [. XXREAL_1:def_2_:_
for r, s being ext-real number holds [.r,s.[ = { a where a is Element of ExtREAL : ( r <= a & a < s ) } ;
:: deftheorem defines ]. XXREAL_1:def_3_:_
for r, s being ext-real number holds ].r,s.] = { a where a is Element of ExtREAL : ( r < a & a <= s ) } ;
:: deftheorem defines ]. XXREAL_1:def_4_:_
for r, s being ext-real number holds ].r,s.[ = { a where a is Element of ExtREAL : ( r < a & a < s ) } ;
theorem Th1: :: XXREAL_1:1
for t, r, s being ext-real number holds
( t in [.r,s.] iff ( r <= t & t <= s ) )
proof
let t, r, s be ext-real number ; ::_thesis: ( t in [.r,s.] iff ( r <= t & t <= s ) )
hereby ::_thesis: ( r <= t & t <= s implies t in [.r,s.] )
assume t in [.r,s.] ; ::_thesis: ( r <= t & t <= s )
then ex a being Element of ExtREAL st
( a = t & r <= a & a <= s ) ;
hence ( r <= t & t <= s ) ; ::_thesis: verum
end;
t is Element of ExtREAL by XXREAL_0:def_1;
hence ( r <= t & t <= s implies t in [.r,s.] ) ; ::_thesis: verum
end;
theorem Th2: :: XXREAL_1:2
for t, r, s being ext-real number holds
( t in ].r,s.] iff ( r < t & t <= s ) )
proof
let t, r, s be ext-real number ; ::_thesis: ( t in ].r,s.] iff ( r < t & t <= s ) )
hereby ::_thesis: ( r < t & t <= s implies t in ].r,s.] )
assume t in ].r,s.] ; ::_thesis: ( r < t & t <= s )
then ex a being Element of ExtREAL st
( a = t & r < a & a <= s ) ;
hence ( r < t & t <= s ) ; ::_thesis: verum
end;
t is Element of ExtREAL by XXREAL_0:def_1;
hence ( r < t & t <= s implies t in ].r,s.] ) ; ::_thesis: verum
end;
theorem Th3: :: XXREAL_1:3
for t, r, s being ext-real number holds
( t in [.r,s.[ iff ( r <= t & t < s ) )
proof
let t, r, s be ext-real number ; ::_thesis: ( t in [.r,s.[ iff ( r <= t & t < s ) )
hereby ::_thesis: ( r <= t & t < s implies t in [.r,s.[ )
assume t in [.r,s.[ ; ::_thesis: ( r <= t & t < s )
then ex a being Element of ExtREAL st
( a = t & r <= a & a < s ) ;
hence ( r <= t & t < s ) ; ::_thesis: verum
end;
t is Element of ExtREAL by XXREAL_0:def_1;
hence ( r <= t & t < s implies t in [.r,s.[ ) ; ::_thesis: verum
end;
theorem Th4: :: XXREAL_1:4
for t, r, s being ext-real number holds
( t in ].r,s.[ iff ( r < t & t < s ) )
proof
let t, r, s be ext-real number ; ::_thesis: ( t in ].r,s.[ iff ( r < t & t < s ) )
hereby ::_thesis: ( r < t & t < s implies t in ].r,s.[ )
assume t in ].r,s.[ ; ::_thesis: ( r < t & t < s )
then ex a being Element of ExtREAL st
( a = t & r < a & a < s ) ;
hence ( r < t & t < s ) ; ::_thesis: verum
end;
t is Element of ExtREAL by XXREAL_0:def_1;
hence ( r < t & t < s implies t in ].r,s.[ ) ; ::_thesis: verum
end;
registration
let r, s be ext-real number ;
cluster[.r,s.] -> ext-real-membered ;
coherence
[.r,s.] is ext-real-membered
proof
let x be set ; :: according to MEMBERED:def_2 ::_thesis: ( not x in [.r,s.] or x is ext-real )
assume x in [.r,s.] ; ::_thesis: x is ext-real
then ex a being Element of ExtREAL st
( x = a & r <= a & a <= s ) ;
hence x is ext-real ; ::_thesis: verum
end;
cluster[.r,s.[ -> ext-real-membered ;
coherence
[.r,s.[ is ext-real-membered
proof
let x be set ; :: according to MEMBERED:def_2 ::_thesis: ( not x in [.r,s.[ or x is ext-real )
assume x in [.r,s.[ ; ::_thesis: x is ext-real
then ex a being Element of ExtREAL st
( x = a & r <= a & a < s ) ;
hence x is ext-real ; ::_thesis: verum
end;
cluster].r,s.] -> ext-real-membered ;
coherence
].r,s.] is ext-real-membered
proof
let x be set ; :: according to MEMBERED:def_2 ::_thesis: ( not x in ].r,s.] or x is ext-real )
assume x in ].r,s.] ; ::_thesis: x is ext-real
then ex a being Element of ExtREAL st
( x = a & r < a & a <= s ) ;
hence x is ext-real ; ::_thesis: verum
end;
cluster].r,s.[ -> ext-real-membered ;
coherence
].r,s.[ is ext-real-membered
proof
let x be set ; :: according to MEMBERED:def_2 ::_thesis: ( not x in ].r,s.[ or x is ext-real )
assume x in ].r,s.[ ; ::_thesis: x is ext-real
then ex a being Element of ExtREAL st
( x = a & r < a & a < s ) ;
hence x is ext-real ; ::_thesis: verum
end;
end;
theorem Th5: :: XXREAL_1:5
for x being set
for p, q being ext-real number holds
( not x in [.p,q.] or x in ].p,q.[ or x = p or x = q )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in [.p,q.] or x in ].p,q.[ or x = p or x = q )
let p, q be ext-real number ; ::_thesis: ( not x in [.p,q.] or x in ].p,q.[ or x = p or x = q )
assume A1: x in [.p,q.] ; ::_thesis: ( x in ].p,q.[ or x = p or x = q )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th1;
A3: s <= q by A1, Th1;
A4: ( p = s or p < s ) by A2, XXREAL_0:1;
( s = q or s < q ) by A3, XXREAL_0:1;
hence ( x in ].p,q.[ or x = p or x = q ) by A4, Th4; ::_thesis: verum
end;
theorem Th6: :: XXREAL_1:6
for x being set
for p, q being ext-real number holds
( not x in [.p,q.] or x in ].p,q.] or x = p )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in [.p,q.] or x in ].p,q.] or x = p )
let p, q be ext-real number ; ::_thesis: ( not x in [.p,q.] or x in ].p,q.] or x = p )
assume A1: x in [.p,q.] ; ::_thesis: ( x in ].p,q.] or x = p )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th1;
A3: s <= q by A1, Th1;
( p = s or p < s ) by A2, XXREAL_0:1;
hence ( x in ].p,q.] or x = p ) by A3, Th2; ::_thesis: verum
end;
theorem Th7: :: XXREAL_1:7
for x being set
for p, q being ext-real number holds
( not x in [.p,q.] or x in [.p,q.[ or x = q )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in [.p,q.] or x in [.p,q.[ or x = q )
let p, q be ext-real number ; ::_thesis: ( not x in [.p,q.] or x in [.p,q.[ or x = q )
assume A1: x in [.p,q.] ; ::_thesis: ( x in [.p,q.[ or x = q )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th1;
s <= q by A1, Th1;
then ( q = s or s < q ) by XXREAL_0:1;
hence ( x in [.p,q.[ or x = q ) by A2, Th3; ::_thesis: verum
end;
theorem Th8: :: XXREAL_1:8
for x being set
for p, q being ext-real number holds
( not x in [.p,q.[ or x in ].p,q.[ or x = p )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in [.p,q.[ or x in ].p,q.[ or x = p )
let p, q be ext-real number ; ::_thesis: ( not x in [.p,q.[ or x in ].p,q.[ or x = p )
assume A1: x in [.p,q.[ ; ::_thesis: ( x in ].p,q.[ or x = p )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th3;
A3: s < q by A1, Th3;
( p = s or p < s ) by A2, XXREAL_0:1;
hence ( x in ].p,q.[ or x = p ) by A3, Th4; ::_thesis: verum
end;
theorem Th9: :: XXREAL_1:9
for x being set
for p, q being ext-real number holds
( not x in ].p,q.] or x in ].p,q.[ or x = q )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in ].p,q.] or x in ].p,q.[ or x = q )
let p, q be ext-real number ; ::_thesis: ( not x in ].p,q.] or x in ].p,q.[ or x = q )
assume A1: x in ].p,q.] ; ::_thesis: ( x in ].p,q.[ or x = q )
then reconsider s = x as ext-real number ;
A2: p < s by A1, Th2;
s <= q by A1, Th2;
then ( q = s or s < q ) by XXREAL_0:1;
hence ( x in ].p,q.[ or x = q ) by A2, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:10
for x being set
for p, q being ext-real number holds
( not x in [.p,q.[ or ( x in ].p,q.] & x <> q ) or x = p )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in [.p,q.[ or ( x in ].p,q.] & x <> q ) or x = p )
let p, q be ext-real number ; ::_thesis: ( not x in [.p,q.[ or ( x in ].p,q.] & x <> q ) or x = p )
assume A1: x in [.p,q.[ ; ::_thesis: ( ( x in ].p,q.] & x <> q ) or x = p )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th3;
A3: s < q by A1, Th3;
( p = s or p < s ) by A2, XXREAL_0:1;
hence ( ( x in ].p,q.] & x <> q ) or x = p ) by A3, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:11
for x being set
for p, q being ext-real number holds
( not x in ].p,q.] or ( x in [.p,q.[ & x <> p ) or x = q )
proof
let x be set ; ::_thesis: for p, q being ext-real number holds
( not x in ].p,q.] or ( x in [.p,q.[ & x <> p ) or x = q )
let p, q be ext-real number ; ::_thesis: ( not x in ].p,q.] or ( x in [.p,q.[ & x <> p ) or x = q )
assume A1: x in ].p,q.] ; ::_thesis: ( ( x in [.p,q.[ & x <> p ) or x = q )
then reconsider s = x as ext-real number ;
A2: p < s by A1, Th2;
s <= q by A1, Th2;
then ( q = s or s < q ) by XXREAL_0:1;
hence ( ( x in [.p,q.[ & x <> p ) or x = q ) by A2, Th3; ::_thesis: verum
end;
theorem Th12: :: XXREAL_1:12
for x being set
for p, q being ext-real number st x in ].p,q.] holds
( x in [.p,q.] & x <> p )
proof
let x be set ; ::_thesis: for p, q being ext-real number st x in ].p,q.] holds
( x in [.p,q.] & x <> p )
let p, q be ext-real number ; ::_thesis: ( x in ].p,q.] implies ( x in [.p,q.] & x <> p ) )
assume A1: x in ].p,q.] ; ::_thesis: ( x in [.p,q.] & x <> p )
then reconsider s = x as ext-real number ;
A2: p < s by A1, Th2;
s <= q by A1, Th2;
hence ( x in [.p,q.] & x <> p ) by A2, Th1; ::_thesis: verum
end;
theorem Th13: :: XXREAL_1:13
for x being set
for p, q being ext-real number st x in [.p,q.[ holds
( x in [.p,q.] & x <> q )
proof
let x be set ; ::_thesis: for p, q being ext-real number st x in [.p,q.[ holds
( x in [.p,q.] & x <> q )
let p, q be ext-real number ; ::_thesis: ( x in [.p,q.[ implies ( x in [.p,q.] & x <> q ) )
assume A1: x in [.p,q.[ ; ::_thesis: ( x in [.p,q.] & x <> q )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th3;
s < q by A1, Th3;
hence ( x in [.p,q.] & x <> q ) by A2, Th1; ::_thesis: verum
end;
theorem Th14: :: XXREAL_1:14
for x being set
for p, q being ext-real number st x in ].p,q.[ holds
( x in [.p,q.[ & x <> p )
proof
let x be set ; ::_thesis: for p, q being ext-real number st x in ].p,q.[ holds
( x in [.p,q.[ & x <> p )
let p, q be ext-real number ; ::_thesis: ( x in ].p,q.[ implies ( x in [.p,q.[ & x <> p ) )
assume A1: x in ].p,q.[ ; ::_thesis: ( x in [.p,q.[ & x <> p )
then reconsider s = x as ext-real number ;
A2: p < s by A1, Th4;
s < q by A1, Th4;
hence ( x in [.p,q.[ & x <> p ) by A2, Th3; ::_thesis: verum
end;
theorem Th15: :: XXREAL_1:15
for x being set
for p, q being ext-real number st x in ].p,q.[ holds
( x in ].p,q.] & x <> q )
proof
let x be set ; ::_thesis: for p, q being ext-real number st x in ].p,q.[ holds
( x in ].p,q.] & x <> q )
let p, q be ext-real number ; ::_thesis: ( x in ].p,q.[ implies ( x in ].p,q.] & x <> q ) )
assume A1: x in ].p,q.[ ; ::_thesis: ( x in ].p,q.] & x <> q )
then reconsider s = x as ext-real number ;
A2: p < s by A1, Th4;
s < q by A1, Th4;
hence ( x in ].p,q.] & x <> q ) by A2, Th2; ::_thesis: verum
end;
theorem Th16: :: XXREAL_1:16
for x being set
for p, q being ext-real number st x in ].p,q.[ holds
( x in [.p,q.] & x <> p & x <> q )
proof
let x be set ; ::_thesis: for p, q being ext-real number st x in ].p,q.[ holds
( x in [.p,q.] & x <> p & x <> q )
let p, q be ext-real number ; ::_thesis: ( x in ].p,q.[ implies ( x in [.p,q.] & x <> p & x <> q ) )
assume A1: x in ].p,q.[ ; ::_thesis: ( x in [.p,q.] & x <> p & x <> q )
then x in ].p,q.] by Th15;
hence ( x in [.p,q.] & x <> p & x <> q ) by A1, Th12, Th15; ::_thesis: verum
end;
theorem Th17: :: XXREAL_1:17
for r being ext-real number holds [.r,r.] = {r}
proof
let r be ext-real number ; ::_thesis: [.r,r.] = {r}
let s be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not s in [.r,r.] or s in {r} ) & ( not s in {r} or s in [.r,r.] ) )
thus ( s in [.r,r.] implies s in {r} ) ::_thesis: ( not s in {r} or s in [.r,r.] )
proof
assume s in [.r,r.] ; ::_thesis: s in {r}
then ex a being Element of ExtREAL st
( s = a & r <= a & a <= r ) ;
then s = r by XXREAL_0:1;
hence s in {r} by TARSKI:def_1; ::_thesis: verum
end;
assume s in {r} ; ::_thesis: s in [.r,r.]
then A1: s = r by TARSKI:def_1;
reconsider s = s as Element of ExtREAL by XXREAL_0:def_1;
s <= s ;
hence s in [.r,r.] by A1; ::_thesis: verum
end;
theorem Th18: :: XXREAL_1:18
for r being ext-real number holds [.r,r.[ = {}
proof
let r be ext-real number ; ::_thesis: [.r,r.[ = {}
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,r.[ or p in {} ) & ( not p in {} or p in [.r,r.[ ) )
for p being ext-real number holds not p in [.r,r.[
proof
given p being ext-real number such that A1: p in [.r,r.[ ; ::_thesis: contradiction
ex a being Element of ExtREAL st
( p = a & r <= a & a < r ) by A1;
hence contradiction ; ::_thesis: verum
end;
hence ( ( not p in [.r,r.[ or p in {} ) & ( not p in {} or p in [.r,r.[ ) ) ; ::_thesis: verum
end;
theorem Th19: :: XXREAL_1:19
for r being ext-real number holds ].r,r.] = {}
proof
let r be ext-real number ; ::_thesis: ].r,r.] = {}
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,r.] or p in {} ) & ( not p in {} or p in ].r,r.] ) )
thus ( p in ].r,r.] implies p in {} ) ::_thesis: ( not p in {} or p in ].r,r.] )
proof
assume p in ].r,r.] ; ::_thesis: p in {}
then ex a being Element of ExtREAL st
( p = a & r < a & a <= r ) ;
hence p in {} ; ::_thesis: verum
end;
thus ( not p in {} or p in ].r,r.] ) ; ::_thesis: verum
end;
theorem Th20: :: XXREAL_1:20
for r being ext-real number holds ].r,r.[ = {}
proof
let r be ext-real number ; ::_thesis: ].r,r.[ = {}
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,r.[ or p in {} ) & ( not p in {} or p in ].r,r.[ ) )
thus ( p in ].r,r.[ implies p in {} ) ::_thesis: ( not p in {} or p in ].r,r.[ )
proof
assume p in ].r,r.[ ; ::_thesis: p in {}
then ex a being Element of ExtREAL st
( p = a & r < a & a < r ) ;
hence p in {} ; ::_thesis: verum
end;
thus ( not p in {} or p in ].r,r.[ ) ; ::_thesis: verum
end;
registration
let r be ext-real number ;
cluster[.r,r.] -> non empty ;
coherence
not [.r,r.] is empty
proof
[.r,r.] = {r} by Th17;
hence not [.r,r.] is empty ; ::_thesis: verum
end;
cluster[.r,r.[ -> empty ;
coherence
[.r,r.[ is empty by Th18;
cluster].r,r.] -> empty ;
coherence
].r,r.] is empty by Th19;
cluster].r,r.[ -> empty ;
coherence
].r,r.[ is empty by Th20;
end;
theorem Th21: :: XXREAL_1:21
for p, q being ext-real number holds ].p,q.[ c= ].p,q.]
proof
let p, q be ext-real number ; ::_thesis: ].p,q.[ c= ].p,q.]
let s be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not s in ].p,q.[ or s in ].p,q.] )
assume A1: s in ].p,q.[ ; ::_thesis: s in ].p,q.]
then A2: p < s by Th4;
s < q by A1, Th4;
hence s in ].p,q.] by A2, Th2; ::_thesis: verum
end;
theorem Th22: :: XXREAL_1:22
for p, q being ext-real number holds ].p,q.[ c= [.p,q.[
proof
let p, q be ext-real number ; ::_thesis: ].p,q.[ c= [.p,q.[
let s be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not s in ].p,q.[ or s in [.p,q.[ )
assume A1: s in ].p,q.[ ; ::_thesis: s in [.p,q.[
then A2: p < s by Th4;
s < q by A1, Th4;
hence s in [.p,q.[ by A2, Th3; ::_thesis: verum
end;
theorem Th23: :: XXREAL_1:23
for p, q being ext-real number holds ].p,q.] c= [.p,q.]
proof
let p, q be ext-real number ; ::_thesis: ].p,q.] c= [.p,q.]
let s be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not s in ].p,q.] or s in [.p,q.] )
assume A1: s in ].p,q.] ; ::_thesis: s in [.p,q.]
then A2: p < s by Th2;
s <= q by A1, Th2;
hence s in [.p,q.] by A2, Th1; ::_thesis: verum
end;
theorem Th24: :: XXREAL_1:24
for p, q being ext-real number holds [.p,q.[ c= [.p,q.]
proof
let p, q be ext-real number ; ::_thesis: [.p,q.[ c= [.p,q.]
let s be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not s in [.p,q.[ or s in [.p,q.] )
assume A1: s in [.p,q.[ ; ::_thesis: s in [.p,q.]
then A2: p <= s by Th3;
s < q by A1, Th3;
hence s in [.p,q.] by A2, Th1; ::_thesis: verum
end;
theorem Th25: :: XXREAL_1:25
for p, q being ext-real number holds ].p,q.[ c= [.p,q.]
proof
let p, q be ext-real number ; ::_thesis: ].p,q.[ c= [.p,q.]
A1: ].p,q.[ c= [.p,q.[ by Th22;
[.p,q.[ c= [.p,q.] by Th24;
hence ].p,q.[ c= [.p,q.] by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem :: XXREAL_1:26
for p, q being ext-real number st p <= q holds
].q,p.] = {}
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies ].q,p.] = {} )
assume A1: p <= q ; ::_thesis: ].q,p.] = {}
assume ].q,p.] <> {} ; ::_thesis: contradiction
then consider r being ext-real number such that
A2: r in ].q,p.] by MEMBERED:8;
A3: q < r by A2, Th2;
r <= p by A2, Th2;
hence contradiction by A1, A3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th27: :: XXREAL_1:27
for p, q being ext-real number st p <= q holds
[.q,p.[ = {}
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.q,p.[ = {} )
assume A1: p <= q ; ::_thesis: [.q,p.[ = {}
assume [.q,p.[ <> {} ; ::_thesis: contradiction
then consider r being ext-real number such that
A2: r in [.q,p.[ by MEMBERED:8;
A3: q <= r by A2, Th3;
r < p by A2, Th3;
hence contradiction by A1, A3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th28: :: XXREAL_1:28
for p, q being ext-real number st p <= q holds
].q,p.[ = {}
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies ].q,p.[ = {} )
assume p <= q ; ::_thesis: ].q,p.[ = {}
then [.q,p.[ = {} by Th27;
hence ].q,p.[ = {} by Th22, XBOOLE_1:3; ::_thesis: verum
end;
theorem Th29: :: XXREAL_1:29
for p, q being ext-real number st p < q holds
[.q,p.] = {}
proof
let p, q be ext-real number ; ::_thesis: ( p < q implies [.q,p.] = {} )
assume A1: p < q ; ::_thesis: [.q,p.] = {}
assume [.q,p.] <> {} ; ::_thesis: contradiction
then consider r being ext-real number such that
A2: r in [.q,p.] by MEMBERED:8;
A3: q <= r by A2, Th1;
r <= p by A2, Th1;
hence contradiction by A1, A3, XXREAL_0:2; ::_thesis: verum
end;
theorem :: XXREAL_1:30
for r, s being ext-real number st r <= s holds
not [.r,s.] is empty by Th1;
theorem :: XXREAL_1:31
for p, q being ext-real number st p < q holds
not [.p,q.[ is empty by Th3;
theorem :: XXREAL_1:32
for p, q being ext-real number st p < q holds
not ].p,q.] is empty by Th2;
theorem :: XXREAL_1:33
for p, q being ext-real number st p < q holds
not ].p,q.[ is empty
proof
let p, q be ext-real number ; ::_thesis: ( p < q implies not ].p,q.[ is empty )
assume p < q ; ::_thesis: not ].p,q.[ is empty
then ex s being ext-real number st
( p < s & s < q ) by XREAL_1:227;
hence not ].p,q.[ is empty by Th4; ::_thesis: verum
end;
theorem Th34: :: XXREAL_1:34
for p, r, s, q being ext-real number st p <= r & s <= q holds
[.r,s.] c= [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies [.r,s.] c= [.p,q.] )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: [.r,s.] c= [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in [.p,q.] )
assume A3: t in [.r,s.] ; ::_thesis: t in [.p,q.]
then A4: r <= t by Th1;
A5: t <= s by A3, Th1;
A6: p <= t by A1, A4, XXREAL_0:2;
t <= q by A2, A5, XXREAL_0:2;
hence t in [.p,q.] by A6, Th1; ::_thesis: verum
end;
theorem Th35: :: XXREAL_1:35
for p, r, s, q being ext-real number st p <= r & s <= q holds
[.r,s.[ c= [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies [.r,s.[ c= [.p,q.] )
A1: [.r,s.[ c= [.r,s.] by Th24;
assume that
A2: p <= r and
A3: s <= q ; ::_thesis: [.r,s.[ c= [.p,q.]
[.r,s.] c= [.p,q.] by A2, A3, Th34;
hence [.r,s.[ c= [.p,q.] by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th36: :: XXREAL_1:36
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.] c= [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.] c= [.p,q.] )
A1: ].r,s.] c= [.r,s.] by Th23;
assume that
A2: p <= r and
A3: s <= q ; ::_thesis: ].r,s.] c= [.p,q.]
[.r,s.] c= [.p,q.] by A2, A3, Th34;
hence ].r,s.] c= [.p,q.] by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th37: :: XXREAL_1:37
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.[ c= [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.[ c= [.p,q.] )
A1: ].r,s.[ c= [.r,s.] by Th25;
assume that
A2: p <= r and
A3: s <= q ; ::_thesis: ].r,s.[ c= [.p,q.]
[.r,s.] c= [.p,q.] by A2, A3, Th34;
hence ].r,s.[ c= [.p,q.] by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th38: :: XXREAL_1:38
for p, r, s, q being ext-real number st p <= r & s <= q holds
[.r,s.[ c= [.p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies [.r,s.[ c= [.p,q.[ )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: [.r,s.[ c= [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.[ or t in [.p,q.[ )
assume A3: t in [.r,s.[ ; ::_thesis: t in [.p,q.[
then A4: r <= t by Th3;
A5: t < s by A3, Th3;
A6: p <= t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in [.p,q.[ by A6, Th3; ::_thesis: verum
end;
theorem Th39: :: XXREAL_1:39
for p, r, s, q being ext-real number st p < r & s <= q holds
[.r,s.] c= ].p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p < r & s <= q implies [.r,s.] c= ].p,q.] )
assume that
A1: p < r and
A2: s <= q ; ::_thesis: [.r,s.] c= ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in ].p,q.] )
assume A3: t in [.r,s.] ; ::_thesis: t in ].p,q.]
then A4: r <= t by Th1;
A5: t <= s by A3, Th1;
A6: p < t by A1, A4, XXREAL_0:2;
t <= q by A2, A5, XXREAL_0:2;
hence t in ].p,q.] by A6, Th2; ::_thesis: verum
end;
theorem Th40: :: XXREAL_1:40
for p, r, s, q being ext-real number st p < r & s <= q holds
[.r,s.[ c= ].p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p < r & s <= q implies [.r,s.[ c= ].p,q.] )
A1: [.r,s.[ c= [.r,s.] by Th24;
assume that
A2: p < r and
A3: s <= q ; ::_thesis: [.r,s.[ c= ].p,q.]
[.r,s.] c= ].p,q.] by A2, A3, Th39;
hence [.r,s.[ c= ].p,q.] by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th41: :: XXREAL_1:41
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.[ c= ].p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.[ c= ].p,q.] )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: ].r,s.[ c= ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.[ or t in ].p,q.] )
assume A3: t in ].r,s.[ ; ::_thesis: t in ].p,q.]
then A4: r < t by Th4;
A5: t < s by A3, Th4;
A6: p < t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in ].p,q.] by A6, Th2; ::_thesis: verum
end;
theorem Th42: :: XXREAL_1:42
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.] c= ].p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.] c= ].p,q.] )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: ].r,s.] c= ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.] or t in ].p,q.] )
assume A3: t in ].r,s.] ; ::_thesis: t in ].p,q.]
then A4: r < t by Th2;
A5: t <= s by A3, Th2;
A6: p < t by A1, A4, XXREAL_0:2;
t <= q by A2, A5, XXREAL_0:2;
hence t in ].p,q.] by A6, Th2; ::_thesis: verum
end;
theorem Th43: :: XXREAL_1:43
for p, r, s, q being ext-real number st p <= r & s < q holds
[.r,s.] c= [.p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s < q implies [.r,s.] c= [.p,q.[ )
assume that
A1: p <= r and
A2: s < q ; ::_thesis: [.r,s.] c= [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in [.p,q.[ )
assume A3: t in [.r,s.] ; ::_thesis: t in [.p,q.[
then A4: r <= t by Th1;
A5: t <= s by A3, Th1;
A6: p <= t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in [.p,q.[ by A6, Th3; ::_thesis: verum
end;
theorem Th44: :: XXREAL_1:44
for p, r, s, q being ext-real number st p <= r & s < q holds
].r,s.] c= [.p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s < q implies ].r,s.] c= [.p,q.[ )
A1: ].r,s.] c= [.r,s.] by Th23;
assume that
A2: p <= r and
A3: s < q ; ::_thesis: ].r,s.] c= [.p,q.[
[.r,s.] c= [.p,q.[ by A2, A3, Th43;
hence ].r,s.] c= [.p,q.[ by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th45: :: XXREAL_1:45
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.[ c= [.p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.[ c= [.p,q.[ )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: ].r,s.[ c= [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.[ or t in [.p,q.[ )
assume A3: t in ].r,s.[ ; ::_thesis: t in [.p,q.[
then A4: r < t by Th4;
A5: t < s by A3, Th4;
A6: p <= t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in [.p,q.[ by A6, Th3; ::_thesis: verum
end;
theorem Th46: :: XXREAL_1:46
for p, r, s, q being ext-real number st p <= r & s <= q holds
].r,s.[ c= ].p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies ].r,s.[ c= ].p,q.[ )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: ].r,s.[ c= ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.[ or t in ].p,q.[ )
assume A3: t in ].r,s.[ ; ::_thesis: t in ].p,q.[
then A4: r < t by Th4;
A5: t < s by A3, Th4;
A6: p < t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in ].p,q.[ by A6, Th4; ::_thesis: verum
end;
theorem Th47: :: XXREAL_1:47
for p, r, s, q being ext-real number st p < r & s < q holds
[.r,s.] c= ].p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p < r & s < q implies [.r,s.] c= ].p,q.[ )
assume that
A1: p < r and
A2: s < q ; ::_thesis: [.r,s.] c= ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in ].p,q.[ )
assume A3: t in [.r,s.] ; ::_thesis: t in ].p,q.[
then A4: r <= t by Th1;
A5: t <= s by A3, Th1;
A6: p < t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in ].p,q.[ by A6, Th4; ::_thesis: verum
end;
theorem Th48: :: XXREAL_1:48
for p, r, s, q being ext-real number st p < r & s <= q holds
[.r,s.[ c= ].p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p < r & s <= q implies [.r,s.[ c= ].p,q.[ )
assume that
A1: p < r and
A2: s <= q ; ::_thesis: [.r,s.[ c= ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.[ or t in ].p,q.[ )
assume A3: t in [.r,s.[ ; ::_thesis: t in ].p,q.[
then A4: r <= t by Th3;
A5: t < s by A3, Th3;
A6: p < t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in ].p,q.[ by A6, Th4; ::_thesis: verum
end;
theorem Th49: :: XXREAL_1:49
for p, r, s, q being ext-real number st p <= r & s < q holds
].r,s.] c= ].p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s < q implies ].r,s.] c= ].p,q.[ )
assume that
A1: p <= r and
A2: s < q ; ::_thesis: ].r,s.] c= ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.] or t in ].p,q.[ )
assume A3: t in ].r,s.] ; ::_thesis: t in ].p,q.[
then A4: r < t by Th2;
A5: t <= s by A3, Th2;
A6: p < t by A1, A4, XXREAL_0:2;
t < q by A2, A5, XXREAL_0:2;
hence t in ].p,q.[ by A6, Th4; ::_thesis: verum
end;
theorem Th50: :: XXREAL_1:50
for r, s, p, q being ext-real number st r <= s & [.r,s.] c= [.p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & [.r,s.] c= [.p,q.] implies ( p <= r & s <= q ) )
assume A1: r <= s ; ::_thesis: ( not [.r,s.] c= [.p,q.] or ( p <= r & s <= q ) )
then A2: r in [.r,s.] by Th1;
s in [.r,s.] by A1, Th1;
hence ( not [.r,s.] c= [.p,q.] or ( p <= r & s <= q ) ) by A2, Th1; ::_thesis: verum
end;
theorem Th51: :: XXREAL_1:51
for r, s, p, q being ext-real number st r < s & ].r,s.[ c= [.p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.[ c= [.p,q.] implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.[ c= [.p,q.] ; ::_thesis: ( p <= r & s <= q )
now__::_thesis:_for_t_being_ext-real_number_st_r_<_t_&_t_<_s_holds_
p_<=_t
let t be ext-real number ; ::_thesis: ( r < t & t < s implies p <= t )
assume that
A3: r < t and
A4: t < s ; ::_thesis: p <= t
t in ].r,s.[ by A3, A4, Th4;
hence p <= t by A2, Th1; ::_thesis: verum
end;
hence p <= r by A1, XREAL_1:228; ::_thesis: s <= q
now__::_thesis:_for_t_being_ext-real_number_st_r_<_t_&_t_<_s_holds_
t_<=_q
let t be ext-real number ; ::_thesis: ( r < t & t < s implies t <= q )
assume that
A5: r < t and
A6: t < s ; ::_thesis: t <= q
t in ].r,s.[ by A5, A6, Th4;
hence t <= q by A2, Th1; ::_thesis: verum
end;
hence s <= q by A1, XREAL_1:229; ::_thesis: verum
end;
theorem Th52: :: XXREAL_1:52
for r, s, p, q being ext-real number st r < s & [.r,s.[ c= [.p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & [.r,s.[ c= [.p,q.] implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: [.r,s.[ c= [.p,q.] ; ::_thesis: ( p <= r & s <= q )
].r,s.[ c= [.r,s.[ by Th22;
then ].r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th53: :: XXREAL_1:53
for r, s, p, q being ext-real number st r < s & ].r,s.] c= [.p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.] c= [.p,q.] implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.] c= [.p,q.] ; ::_thesis: ( p <= r & s <= q )
].r,s.[ c= ].r,s.] by Th21;
then ].r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th54: :: XXREAL_1:54
for r, s, p, q being ext-real number st r <= s & [.r,s.] c= [.p,q.[ holds
( p <= r & s < q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & [.r,s.] c= [.p,q.[ implies ( p <= r & s < q ) )
assume that
A1: r <= s and
A2: [.r,s.] c= [.p,q.[ ; ::_thesis: ( p <= r & s < q )
[.p,q.[ c= [.p,q.] by Th24;
then [.r,s.] c= [.p,q.] by A2, XBOOLE_1:1;
hence p <= r by A1, Th50; ::_thesis: s < q
s in [.r,s.] by A1, Th1;
hence s < q by A2, Th3; ::_thesis: verum
end;
theorem Th55: :: XXREAL_1:55
for r, s, p, q being ext-real number st r < s & [.r,s.[ c= [.p,q.[ holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & [.r,s.[ c= [.p,q.[ implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: [.r,s.[ c= [.p,q.[ ; ::_thesis: ( p <= r & s <= q )
].r,s.[ c= [.r,s.[ by Th22;
then A3: ].r,s.[ c= [.p,q.[ by A2, XBOOLE_1:1;
[.p,q.[ c= [.p,q.] by Th24;
then ].r,s.[ c= [.p,q.] by A3, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th56: :: XXREAL_1:56
for r, s, p, q being ext-real number st r < s & ].r,s.[ c= [.p,q.[ holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.[ c= [.p,q.[ implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.[ c= [.p,q.[ ; ::_thesis: ( p <= r & s <= q )
[.p,q.[ c= [.p,q.] by Th24;
then ].r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th57: :: XXREAL_1:57
for r, s, p, q being ext-real number st r < s & ].r,s.] c= [.p,q.[ holds
( p <= r & s < q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.] c= [.p,q.[ implies ( p <= r & s < q ) )
assume that
A1: r < s and
A2: ].r,s.] c= [.p,q.[ ; ::_thesis: ( p <= r & s < q )
[.p,q.[ c= [.p,q.] by Th24;
then ].r,s.] c= [.p,q.] by A2, XBOOLE_1:1;
hence p <= r by A1, Th53; ::_thesis: s < q
s in ].r,s.] by A1, Th2;
hence s < q by A2, Th3; ::_thesis: verum
end;
theorem Th58: :: XXREAL_1:58
for r, s, p, q being ext-real number st r <= s & [.r,s.] c= ].p,q.] holds
( p < r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & [.r,s.] c= ].p,q.] implies ( p < r & s <= q ) )
assume that
A1: r <= s and
A2: [.r,s.] c= ].p,q.] ; ::_thesis: ( p < r & s <= q )
].p,q.] c= [.p,q.] by Th23;
then A3: [.r,s.] c= [.p,q.] by A2, XBOOLE_1:1;
r in [.r,s.] by A1, Th1;
hence p < r by A2, Th2; ::_thesis: s <= q
thus s <= q by A1, A3, Th50; ::_thesis: verum
end;
theorem Th59: :: XXREAL_1:59
for r, s, p, q being ext-real number st r < s & ].r,s.[ c= ].p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.[ c= ].p,q.] implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.[ c= ].p,q.] ; ::_thesis: ( p <= r & s <= q )
].p,q.] c= [.p,q.] by Th23;
then ].r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th60: :: XXREAL_1:60
for r, s, p, q being ext-real number st r < s & [.r,s.[ c= ].p,q.] holds
( p < r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & [.r,s.[ c= ].p,q.] implies ( p < r & s <= q ) )
assume that
A1: r < s and
A2: [.r,s.[ c= ].p,q.] ; ::_thesis: ( p < r & s <= q )
].p,q.] c= [.p,q.] by Th23;
then A3: [.r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
r in [.r,s.[ by A1, Th3;
hence p < r by A2, Th2; ::_thesis: s <= q
thus s <= q by A1, A3, Th52; ::_thesis: verum
end;
theorem Th61: :: XXREAL_1:61
for r, s, p, q being ext-real number st r < s & ].r,s.] c= ].p,q.] holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.] c= ].p,q.] implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.] c= ].p,q.] ; ::_thesis: ( p <= r & s <= q )
].r,s.[ c= ].r,s.] by Th21;
then A3: ].r,s.[ c= ].p,q.] by A2, XBOOLE_1:1;
].p,q.] c= [.p,q.] by Th23;
then ].r,s.[ c= [.p,q.] by A3, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th62: :: XXREAL_1:62
for r, s, p, q being ext-real number st r <= s & [.r,s.] c= ].p,q.[ holds
( p < r & s < q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & [.r,s.] c= ].p,q.[ implies ( p < r & s < q ) )
assume that
A1: r <= s and
A2: [.r,s.] c= ].p,q.[ ; ::_thesis: ( p < r & s < q )
r in [.r,s.] by A1, Th1;
hence p < r by A2, Th4; ::_thesis: s < q
s in [.r,s.] by A1, Th1;
hence s < q by A2, Th4; ::_thesis: verum
end;
theorem Th63: :: XXREAL_1:63
for r, s, p, q being ext-real number st r < s & ].r,s.[ c= ].p,q.[ holds
( p <= r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.[ c= ].p,q.[ implies ( p <= r & s <= q ) )
assume that
A1: r < s and
A2: ].r,s.[ c= ].p,q.[ ; ::_thesis: ( p <= r & s <= q )
].p,q.[ c= [.p,q.] by Th25;
then ].r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
hence ( p <= r & s <= q ) by A1, Th51; ::_thesis: verum
end;
theorem Th64: :: XXREAL_1:64
for r, s, p, q being ext-real number st r < s & [.r,s.[ c= ].p,q.[ holds
( p < r & s <= q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & [.r,s.[ c= ].p,q.[ implies ( p < r & s <= q ) )
assume that
A1: r < s and
A2: [.r,s.[ c= ].p,q.[ ; ::_thesis: ( p < r & s <= q )
].p,q.[ c= [.p,q.] by Th25;
then A3: [.r,s.[ c= [.p,q.] by A2, XBOOLE_1:1;
r in [.r,s.[ by A1, Th3;
hence p < r by A2, Th4; ::_thesis: s <= q
thus s <= q by A1, A3, Th52; ::_thesis: verum
end;
theorem Th65: :: XXREAL_1:65
for r, s, p, q being ext-real number st r < s & ].r,s.] c= ].p,q.[ holds
( p <= r & s < q )
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.] c= ].p,q.[ implies ( p <= r & s < q ) )
assume that
A1: r < s and
A2: ].r,s.] c= ].p,q.[ ; ::_thesis: ( p <= r & s < q )
].p,q.[ c= [.p,q.] by Th25;
then ].r,s.] c= [.p,q.] by A2, XBOOLE_1:1;
hence p <= r by A1, Th53; ::_thesis: s < q
s in ].r,s.] by A1, Th2;
hence s < q by A2, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:66
for p, q, r, s being ext-real number st p <= q & [.p,q.] = [.r,s.] holds
( p = r & q = s )
proof
let p, q, r, s be ext-real number ; ::_thesis: ( p <= q & [.p,q.] = [.r,s.] implies ( p = r & q = s ) )
assume that
A1: p <= q and
A2: [.p,q.] = [.r,s.] ; ::_thesis: ( p = r & q = s )
A3: r <= p by A1, A2, Th50;
A4: q <= s by A1, A2, Th50;
r <= q by A1, A3, XXREAL_0:2;
then A5: r <= s by A4, XXREAL_0:2;
then A6: p <= r by A2, Th50;
s <= q by A2, A5, Th50;
hence ( p = r & q = s ) by A3, A4, A6, XXREAL_0:1; ::_thesis: verum
end;
theorem :: XXREAL_1:67
for p, q, r, s being ext-real number st p < q & ].p,q.[ = ].r,s.[ holds
( p = r & q = s )
proof
let p, q, r, s be ext-real number ; ::_thesis: ( p < q & ].p,q.[ = ].r,s.[ implies ( p = r & q = s ) )
assume that
A1: p < q and
A2: ].p,q.[ = ].r,s.[ ; ::_thesis: ( p = r & q = s )
A3: r <= p by A1, A2, Th63;
A4: q <= s by A1, A2, Th63;
r < q by A1, A3, XXREAL_0:2;
then A5: r < s by A4, XXREAL_0:2;
then A6: p <= r by A2, Th63;
s <= q by A2, A5, Th63;
hence ( p = r & q = s ) by A3, A4, A6, XXREAL_0:1; ::_thesis: verum
end;
theorem :: XXREAL_1:68
for p, q, r, s being ext-real number st p < q & ].p,q.] = ].r,s.] holds
( p = r & q = s )
proof
let p, q, r, s be ext-real number ; ::_thesis: ( p < q & ].p,q.] = ].r,s.] implies ( p = r & q = s ) )
assume that
A1: p < q and
A2: ].p,q.] = ].r,s.] ; ::_thesis: ( p = r & q = s )
A3: r <= p by A1, A2, Th61;
A4: q <= s by A1, A2, Th61;
r < q by A1, A3, XXREAL_0:2;
then A5: r < s by A4, XXREAL_0:2;
then A6: p <= r by A2, Th61;
s <= q by A2, A5, Th61;
hence ( p = r & q = s ) by A3, A4, A6, XXREAL_0:1; ::_thesis: verum
end;
theorem :: XXREAL_1:69
for p, q, r, s being ext-real number st p < q & [.p,q.[ = [.r,s.[ holds
( p = r & q = s )
proof
let p, q, r, s be ext-real number ; ::_thesis: ( p < q & [.p,q.[ = [.r,s.[ implies ( p = r & q = s ) )
assume that
A1: p < q and
A2: [.p,q.[ = [.r,s.[ ; ::_thesis: ( p = r & q = s )
A3: r <= p by A1, A2, Th55;
A4: q <= s by A1, A2, Th55;
r < q by A1, A3, XXREAL_0:2;
then A5: r < s by A4, XXREAL_0:2;
then A6: p <= r by A2, Th55;
s <= q by A2, A5, Th55;
hence ( p = r & q = s ) by A3, A4, A6, XXREAL_0:1; ::_thesis: verum
end;
theorem :: XXREAL_1:70
for r, s, p, q being ext-real number st r <= s holds
[.r,s.] <> ].p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] <> ].p,q.] )
assume that
A1: r <= s and
A2: [.r,s.] = ].p,q.] ; ::_thesis: contradiction
now__::_thesis:_not_r_in_].p,q.]
assume r in ].p,q.] ; ::_thesis: contradiction
then A3: p < r by Th2;
s <= q by A1, A2, Th58;
then r <= q by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th53; ::_thesis: verum
end;
hence contradiction by A1, A2, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:71
for r, s, p, q being ext-real number st r <= s holds
[.r,s.] <> [.p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] <> [.p,q.[ )
assume that
A1: r <= s and
A2: [.r,s.] = [.p,q.[ ; ::_thesis: contradiction
now__::_thesis:_not_s_in_[.p,q.[
assume s in [.p,q.[ ; ::_thesis: contradiction
then A3: s < q by Th3;
p <= r by A1, A2, Th54;
then p <= s by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th52; ::_thesis: verum
end;
hence contradiction by A1, A2, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:72
for r, s, p, q being ext-real number st r <= s holds
[.r,s.] <> ].p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] <> ].p,q.[ )
assume that
A1: r <= s and
A2: [.r,s.] = ].p,q.[ ; ::_thesis: contradiction
now__::_thesis:_not_s_in_].p,q.[
assume s in ].p,q.[ ; ::_thesis: contradiction
then A3: s < q by Th4;
p <= r by A1, A2, Th62;
then p <= s by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th51; ::_thesis: verum
end;
hence contradiction by A1, A2, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:73
for r, s, p, q being ext-real number st r < s holds
[.r,s.[ <> [.p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies [.r,s.[ <> [.p,q.] )
assume that
A1: r < s and
A2: [.r,s.[ = [.p,q.] ; ::_thesis: contradiction
A3: not s in [.r,s.[ by Th3;
p <= r by A1, A2, Th52;
then A4: p <= s by A1, XXREAL_0:2;
s <= q by A1, A2, Th52;
hence contradiction by A2, A3, A4, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:74
for r, s, p, q being ext-real number st r < s holds
[.r,s.[ <> ].p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies [.r,s.[ <> ].p,q.] )
assume that
A1: r < s and
A2: [.r,s.[ = ].p,q.] ; ::_thesis: contradiction
A3: not s in [.r,s.[ by Th3;
p <= r by A1, A2, Th60;
then A4: p < s by A1, XXREAL_0:2;
s <= q by A1, A2, Th60;
hence contradiction by A2, A3, A4, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:75
for r, s, p, q being ext-real number st r < s holds
[.r,s.[ <> ].p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies [.r,s.[ <> ].p,q.[ )
assume that
A1: r < s and
A2: [.r,s.[ = ].p,q.[ ; ::_thesis: contradiction
now__::_thesis:_not_r_in_].p,q.[
assume r in ].p,q.[ ; ::_thesis: contradiction
then A3: p < r by Th4;
s <= q by A1, A2, Th64;
then r < q by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th56; ::_thesis: verum
end;
hence contradiction by A1, A2, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:76
for r, s, p, q being ext-real number st r < s holds
].r,s.] <> [.p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.] <> [.p,q.] )
assume that
A1: r < s and
A2: ].r,s.] = [.p,q.] ; ::_thesis: contradiction
A3: not r in ].r,s.] by Th2;
A4: p <= r by A1, A2, Th53;
s <= q by A1, A2, Th53;
then r <= q by A1, XXREAL_0:2;
hence contradiction by A2, A3, A4, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:77
for r, s, p, q being ext-real number st r < s holds
].r,s.] <> [.p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.] <> [.p,q.[ )
assume that
A1: r < s and
A2: ].r,s.] = [.p,q.[ ; ::_thesis: contradiction
A3: not r in ].r,s.] by Th2;
A4: p <= r by A1, A2, Th57;
s <= q by A1, A2, Th57;
then r < q by A1, XXREAL_0:2;
hence contradiction by A2, A3, A4, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:78
for r, s, p, q being ext-real number st r < s holds
].r,s.] <> ].p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.] <> ].p,q.[ )
assume that
A1: r < s and
A2: ].r,s.] = ].p,q.[ ; ::_thesis: contradiction
now__::_thesis:_not_s_in_].p,q.[
assume s in ].p,q.[ ; ::_thesis: contradiction
then A3: s < q by Th4;
p <= r by A1, A2, Th65;
then p <= s by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th59; ::_thesis: verum
end;
hence contradiction by A1, A2, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:79
for r, s, p, q being ext-real number st r < s holds
].r,s.[ <> [.p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.[ <> [.p,q.] )
assume that
A1: r < s and
A2: ].r,s.[ = [.p,q.] ; ::_thesis: contradiction
A3: not r in ].r,s.[ by Th4;
A4: p <= r by A1, A2, Th51;
s <= q by A1, A2, Th51;
then r <= q by A1, XXREAL_0:2;
hence contradiction by A2, A3, A4, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:80
for r, s, p, q being ext-real number st r < s holds
].r,s.[ <> ].p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.[ <> ].p,q.] )
assume that
A1: r < s and
A2: ].r,s.[ = ].p,q.] ; ::_thesis: contradiction
A3: not s in ].r,s.[ by Th4;
p <= r by A1, A2, Th59;
then A4: p < s by A1, XXREAL_0:2;
s <= q by A1, A2, Th59;
hence contradiction by A2, A3, A4, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:81
for r, s, p, q being ext-real number st r < s holds
].r,s.[ <> [.p,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s implies ].r,s.[ <> [.p,q.[ )
assume that
A1: r < s and
A2: ].r,s.[ = [.p,q.[ ; ::_thesis: contradiction
A3: not r in ].r,s.[ by Th4;
A4: p <= r by A1, A2, Th56;
s <= q by A1, A2, Th56;
then r < q by A1, XXREAL_0:2;
hence contradiction by A2, A3, A4, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:82
for r, s, p, q being ext-real number st r <= s & [.r,s.] c< [.p,q.] & not p < r holds
s < q
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & [.r,s.] c< [.p,q.] & not p < r implies s < q )
assume A1: r <= s ; ::_thesis: ( not [.r,s.] c< [.p,q.] or p < r or s < q )
assume A2: [.r,s.] c< [.p,q.] ; ::_thesis: ( p < r or s < q )
then A3: [.r,s.] c= [.p,q.] by XBOOLE_0:def_8;
then A4: p <= r by A1, Th50;
A5: s <= q by A1, A3, Th50;
( p <> r or s <> q ) by A2;
hence ( p < r or s < q ) by A4, A5, XXREAL_0:1; ::_thesis: verum
end;
theorem :: XXREAL_1:83
for r, s, p, q being ext-real number st r < s & ].r,s.[ c= [.p,q.] holds
[.r,s.] c= [.p,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & ].r,s.[ c= [.p,q.] implies [.r,s.] c= [.p,q.] )
assume that
A1: r < s and
A2: ].r,s.[ c= [.p,q.] ; ::_thesis: [.r,s.] c= [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in [.p,q.] )
assume A3: t in [.r,s.] ; ::_thesis: t in [.p,q.]
percases ( t in ].r,s.[ or t = r or t = s ) by A3, Th5;
suppose t in ].r,s.[ ; ::_thesis: t in [.p,q.]
hence t in [.p,q.] by A2; ::_thesis: verum
end;
supposeA4: t = r ; ::_thesis: t in [.p,q.]
then A5: p <= t by A1, A2, Th51;
s <= q by A1, A2, Th51;
then t <= q by A1, A4, XXREAL_0:2;
hence t in [.p,q.] by A5, Th1; ::_thesis: verum
end;
supposeA6: t = s ; ::_thesis: t in [.p,q.]
A7: s <= q by A1, A2, Th51;
p <= r by A1, A2, Th51;
then p <= t by A1, A6, XXREAL_0:2;
hence t in [.p,q.] by A6, A7, Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:84
for r, s, p being ext-real number st r < s holds
[.s,p.[ c= ].r,p.[
proof
let r, s, p be ext-real number ; ::_thesis: ( r < s implies [.s,p.[ c= ].r,p.[ )
assume A1: r < s ; ::_thesis: [.s,p.[ c= ].r,p.[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.s,p.[ or t in ].r,p.[ )
assume A2: t in [.s,p.[ ; ::_thesis: t in ].r,p.[
then s <= t by Th3;
then A3: r < t by A1, XXREAL_0:2;
t < p by A2, Th3;
hence t in ].r,p.[ by A3, Th4; ::_thesis: verum
end;
theorem Th85: :: XXREAL_1:85
for s, r being ext-real number st s <= r holds
( [.r,s.] c= {r} & [.r,s.] c= {s} )
proof
let s, r be ext-real number ; ::_thesis: ( s <= r implies ( [.r,s.] c= {r} & [.r,s.] c= {s} ) )
assume A1: s <= r ; ::_thesis: ( [.r,s.] c= {r} & [.r,s.] c= {s} )
thus [.r,s.] c= {r} ::_thesis: [.r,s.] c= {s}
proof
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in {r} )
assume A2: t in [.r,s.] ; ::_thesis: t in {r}
then A3: t <= s by Th1;
A4: r <= t by A2, Th1;
t <= r by A1, A3, XXREAL_0:2;
then r = t by A4, XXREAL_0:1;
hence t in {r} by TARSKI:def_1; ::_thesis: verum
end;
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.] or t in {s} )
assume A5: t in [.r,s.] ; ::_thesis: t in {s}
then r <= t by Th1;
then A6: s <= t by A1, XXREAL_0:2;
t <= s by A5, Th1;
then s = t by A6, XXREAL_0:1;
hence t in {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:86
for r, s being ext-real number holds ].r,s.[ misses {r,s}
proof
let r, s be ext-real number ; ::_thesis: ].r,s.[ misses {r,s}
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.[ or not t in {r,s} )
assume A1: t in ].r,s.[ ; ::_thesis: not t in {r,s}
then A2: r < t by Th4;
t < s by A1, Th4;
hence not t in {r,s} by A2, TARSKI:def_2; ::_thesis: verum
end;
theorem :: XXREAL_1:87
for r, s being ext-real number holds [.r,s.[ misses {s}
proof
let r, s be ext-real number ; ::_thesis: [.r,s.[ misses {s}
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.[ or not t in {s} )
assume t in [.r,s.[ ; ::_thesis: not t in {s}
then t < s by Th3;
hence not t in {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:88
for r, s being ext-real number holds ].r,s.] misses {r}
proof
let r, s be ext-real number ; ::_thesis: ].r,s.] misses {r}
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.] or not t in {r} )
assume t in ].r,s.] ; ::_thesis: not t in {r}
then r < t by Th2;
hence not t in {r} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:89
for s, p, r, q being ext-real number st s <= p holds
[.r,s.] misses ].p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.] misses ].p,q.[ )
assume A1: s <= p ; ::_thesis: [.r,s.] misses ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.] or not t in ].p,q.[ )
assume t in [.r,s.] ; ::_thesis: not t in ].p,q.[
then t <= s by Th1;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:90
for s, p, r, q being ext-real number st s <= p holds
[.r,s.] misses ].p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.] misses ].p,q.] )
assume A1: s <= p ; ::_thesis: [.r,s.] misses ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.] or not t in ].p,q.] )
assume t in [.r,s.] ; ::_thesis: not t in ].p,q.]
then t <= s by Th1;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:91
for s, p, r, q being ext-real number st s <= p holds
].r,s.] misses ].p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.] misses ].p,q.[ )
assume A1: s <= p ; ::_thesis: ].r,s.] misses ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.] or not t in ].p,q.[ )
assume t in ].r,s.] ; ::_thesis: not t in ].p,q.[
then t <= s by Th2;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:92
for s, p, r, q being ext-real number st s <= p holds
].r,s.] misses ].p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.] misses ].p,q.] )
assume A1: s <= p ; ::_thesis: ].r,s.] misses ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.] or not t in ].p,q.] )
assume t in ].r,s.] ; ::_thesis: not t in ].p,q.]
then t <= s by Th2;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:93
for s, p, r, q being ext-real number st s <= p holds
].r,s.[ misses [.p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.[ misses [.p,q.] )
assume A1: s <= p ; ::_thesis: ].r,s.[ misses [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.[ or not t in [.p,q.] )
assume t in ].r,s.[ ; ::_thesis: not t in [.p,q.]
then t < s by Th4;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:94
for s, p, r, q being ext-real number st s <= p holds
].r,s.[ misses [.p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.[ misses [.p,q.[ )
assume A1: s <= p ; ::_thesis: ].r,s.[ misses [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.[ or not t in [.p,q.[ )
assume t in ].r,s.[ ; ::_thesis: not t in [.p,q.[
then t < s by Th4;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:95
for s, p, r, q being ext-real number st s <= p holds
[.r,s.[ misses [.p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.[ misses [.p,q.] )
assume A1: s <= p ; ::_thesis: [.r,s.[ misses [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.[ or not t in [.p,q.] )
assume t in [.r,s.[ ; ::_thesis: not t in [.p,q.]
then t < s by Th3;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:96
for s, p, r, q being ext-real number st s <= p holds
[.r,s.[ misses [.p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.[ misses [.p,q.[ )
assume A1: s <= p ; ::_thesis: [.r,s.[ misses [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.[ or not t in [.p,q.[ )
assume t in [.r,s.[ ; ::_thesis: not t in [.p,q.[
then t < s by Th3;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:97
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.[ c= [.p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.[ c= [.p,q.] )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.[ c= [.p,q.]
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.[ c= [.p,q.]
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.] )
thus t in ].r,s.[ by A4, A5, Th4; ::_thesis: not t in [.p,q.]
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.[ c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.] )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.[ by A7, Th4; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A8, Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:98
for r, p, s, q being ext-real number st r < p & r < s holds
not [.r,s.[ c= [.p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not [.r,s.[ c= [.p,q.] )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not [.r,s.[ c= [.p,q.]
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not [.r,s.[ c= [.p,q.]
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.] )
thus t in [.r,s.[ by A4, A5, Th3; ::_thesis: not t in [.p,q.]
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not [.r,s.[ c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.] )
t < s by A6, A8, XXREAL_0:2;
hence t in [.r,s.[ by A7, Th3; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A8, Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:99
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.] c= [.p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.] c= [.p,q.] )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.] c= [.p,q.]
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.] c= [.p,q.]
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.] )
thus t in ].r,s.] by A4, A5, Th2; ::_thesis: not t in [.p,q.]
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.] c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.] )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.] by A7, Th2; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A8, Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:100
for r, p, s, q being ext-real number st r < p & r <= s holds
not [.r,s.] c= [.p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r <= s implies not [.r,s.] c= [.p,q.] )
assume that
A1: r < p and
A2: r <= s ; ::_thesis: not [.r,s.] c= [.p,q.]
take t = r; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in [.p,q.] )
thus t in [.r,s.] by A2, Th1; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A1, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:101
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.[ c= [.p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.[ c= [.p,q.[ )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.[ c= [.p,q.[
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.[ c= [.p,q.[
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.[ )
thus t in ].r,s.[ by A4, A5, Th4; ::_thesis: not t in [.p,q.[
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.[ c= [.p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.[ )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.[ by A7, Th4; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A8, Th3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:102
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.] c= [.p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.] c= [.p,q.[ )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.] c= [.p,q.[
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.] c= [.p,q.[
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.[ )
thus t in ].r,s.] by A4, A5, Th2; ::_thesis: not t in [.p,q.[
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.] c= [.p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.[ )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.] by A7, Th2; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A8, Th3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:103
for r, p, s, q being ext-real number st r < p & r < s holds
not [.r,s.[ c= [.p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not [.r,s.[ c= [.p,q.[ )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not [.r,s.[ c= [.p,q.[
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not [.r,s.[ c= [.p,q.[
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.[ )
thus t in [.r,s.[ by A4, A5, Th3; ::_thesis: not t in [.p,q.[
t < p by A3, A5, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not [.r,s.[ c= [.p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.[ )
t < s by A6, A8, XXREAL_0:2;
hence t in [.r,s.[ by A7, Th3; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A8, Th3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:104
for r, p, s, q being ext-real number st r < p & r <= s holds
not [.r,s.] c= [.p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r <= s implies not [.r,s.] c= [.p,q.[ )
assume that
A1: r < p and
A2: r <= s ; ::_thesis: not [.r,s.] c= [.p,q.[
take t = r; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in [.p,q.[ )
thus t in [.r,s.] by A2, Th1; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A1, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:105
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.[ c= ].p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.[ c= ].p,q.] )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.[ c= ].p,q.]
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.[ c= ].p,q.]
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.] )
thus t in ].r,s.[ by A4, A5, Th4; ::_thesis: not t in ].p,q.]
t < p by A3, A5, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.[ c= ].p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.] )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.[ by A7, Th4; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A8, Th2; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:106
for r, p, s, q being ext-real number st r <= p & r < s holds
not [.r,s.[ c= ].p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & r < s implies not [.r,s.[ c= ].p,q.] )
assume that
A1: r <= p and
A2: r < s ; ::_thesis: not [.r,s.[ c= ].p,q.]
take t = r; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in ].p,q.] )
thus t in [.r,s.[ by A2, Th3; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A1, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:107
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.] c= ].p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.] c= ].p,q.] )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.] c= ].p,q.]
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.] c= ].p,q.]
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.] )
thus t in ].r,s.] by A4, A5, Th2; ::_thesis: not t in ].p,q.]
t < p by A3, A5, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.] c= ].p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.] )
t <= s by A6, A8, XXREAL_0:2;
hence t in ].r,s.] by A7, Th2; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A8, Th2; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:108
for r, p, s, q being ext-real number st r <= p & r <= s holds
not [.r,s.] c= ].p,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & r <= s implies not [.r,s.] c= ].p,q.] )
assume that
A1: r <= p and
A2: r <= s ; ::_thesis: not [.r,s.] c= ].p,q.]
take r ; :: according to MEMBERED:def_8 ::_thesis: ( r in [.r,s.] & not r in ].p,q.] )
thus r in [.r,s.] by A2, Th1; ::_thesis: not r in ].p,q.]
thus not r in ].p,q.] by A1, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:109
for r, p, s, q being ext-real number st r <= p & r <= s holds
not [.r,s.] c= ].p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & r <= s implies not [.r,s.] c= ].p,q.[ )
assume that
A1: r <= p and
A2: r <= s ; ::_thesis: not [.r,s.] c= ].p,q.[
take r ; :: according to MEMBERED:def_8 ::_thesis: ( r in [.r,s.] & not r in ].p,q.[ )
thus r in [.r,s.] by A2, Th1; ::_thesis: not r in ].p,q.[
thus not r in ].p,q.[ by A1, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:110
for r, p, s, q being ext-real number st r <= p & r < s holds
not [.r,s.[ c= ].p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & r < s implies not [.r,s.[ c= ].p,q.[ )
assume that
A1: r <= p and
A2: r < s ; ::_thesis: not [.r,s.[ c= ].p,q.[
take r ; :: according to MEMBERED:def_8 ::_thesis: ( r in [.r,s.[ & not r in ].p,q.[ )
thus r in [.r,s.[ by A2, Th3; ::_thesis: not r in ].p,q.[
thus not r in ].p,q.[ by A1, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:111
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.] c= ].p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.] c= ].p,q.[ )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.] c= ].p,q.[
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.] c= ].p,q.[
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.[ )
thus t in ].r,s.] by A4, A5, Th2; ::_thesis: not t in ].p,q.[
t < p by A3, A5, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.] c= ].p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.[ )
t <= s by A6, A8, XXREAL_0:2;
hence t in ].r,s.] by A7, Th2; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A8, Th4; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:112
for r, p, s, q being ext-real number st r < p & r < s holds
not ].r,s.[ c= ].p,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & r < s implies not ].r,s.[ c= ].p,q.[ )
assume that
A1: r < p and
A2: r < s ; ::_thesis: not ].r,s.[ c= ].p,q.[
percases ( s <= p or p <= s ) ;
supposeA3: s <= p ; ::_thesis: not ].r,s.[ c= ].p,q.[
consider t being ext-real number such that
A4: r < t and
A5: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.[ )
thus t in ].r,s.[ by A4, A5, Th4; ::_thesis: not t in ].p,q.[
t < p by A3, A5, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
supposeA6: p <= s ; ::_thesis: not ].r,s.[ c= ].p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < p by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.[ )
t < s by A6, A8, XXREAL_0:2;
hence t in ].r,s.[ by A7, Th4; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A8, Th4; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:113
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.[ c= [.p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.[ c= [.p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.[ c= [.p,q.]
percases ( r <= q or q <= r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.[ c= [.p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.[ by A5, Th4; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A4, Th1; ::_thesis: verum
end;
supposeA6: q <= r ; ::_thesis: not ].r,s.[ c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.] )
thus t in ].r,s.[ by A7, A8, Th4; ::_thesis: not t in [.p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:114
for q, s, r, p being ext-real number st q < s & r < s holds
not [.r,s.[ c= [.p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not [.r,s.[ c= [.p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not [.r,s.[ c= [.p,q.]
percases ( r <= q or q <= r ) ;
supposeA3: r <= q ; ::_thesis: not [.r,s.[ c= [.p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in [.r,s.[ by A5, Th3; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A4, Th1; ::_thesis: verum
end;
supposeA6: q <= r ; ::_thesis: not [.r,s.[ c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.] )
thus t in [.r,s.[ by A7, A8, Th3; ::_thesis: not t in [.p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:115
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.] c= [.p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.] c= [.p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.] c= [.p,q.]
percases ( r <= q or q <= r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.] c= [.p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.] by A5, Th2; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A4, Th1; ::_thesis: verum
end;
supposeA6: q <= r ; ::_thesis: not ].r,s.] c= [.p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.] )
thus t in ].r,s.] by A7, A8, Th2; ::_thesis: not t in [.p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:116
for q, s, r, p being ext-real number st q < s & r <= s holds
not [.r,s.] c= [.p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r <= s implies not [.r,s.] c= [.p,q.] )
assume that
A1: q < s and
A2: r <= s ; ::_thesis: not [.r,s.] c= [.p,q.]
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not [.r,s.] c= [.p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in [.p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in [.r,s.] by A5, Th1; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A4, Th1; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not [.r,s.] c= [.p,q.]
take t = r; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in [.p,q.] )
thus t in [.r,s.] by A2, Th1; ::_thesis: not t in [.p,q.]
thus not t in [.p,q.] by A6, Th1; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:117
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.[ c= [.p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.[ c= [.p,q.[ )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.[ c= [.p,q.[
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.[ c= [.p,q.[
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.[ )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.[ by A5, Th4; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A4, Th3; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not ].r,s.[ c= [.p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in [.p,q.[ )
thus t in ].r,s.[ by A7, A8, Th4; ::_thesis: not t in [.p,q.[
q < t by A6, A7, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:118
for q, s, r, p being ext-real number st q <= s & r < s holds
not ].r,s.] c= [.p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q <= s & r < s implies not ].r,s.] c= [.p,q.[ )
assume that
A1: q <= s and
A2: r < s ; ::_thesis: not ].r,s.] c= [.p,q.[
take t = s; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in [.p,q.[ )
thus t in ].r,s.] by A2, Th2; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A1, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:119
for q, s, r, p being ext-real number st q < s & r < s holds
not [.r,s.[ c= [.p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not [.r,s.[ c= [.p,q.[ )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not [.r,s.[ c= [.p,q.[
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not [.r,s.[ c= [.p,q.[
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.[ )
r < t by A3, A4, XXREAL_0:2;
hence t in [.r,s.[ by A5, Th3; ::_thesis: not t in [.p,q.[
thus not t in [.p,q.[ by A4, Th3; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not [.r,s.[ c= [.p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in [.p,q.[ )
thus t in [.r,s.[ by A7, A8, Th3; ::_thesis: not t in [.p,q.[
q < t by A6, A7, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:120
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.[ c= ].p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.[ c= ].p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.[ c= ].p,q.]
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.[ c= ].p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.[ by A5, Th4; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A4, Th2; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not ].r,s.[ c= ].p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.] )
thus t in ].r,s.[ by A7, A8, Th4; ::_thesis: not t in ].p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:121
for q, s, r, p being ext-real number st q < s & r <= s holds
not [.r,s.] c= ].p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r <= s implies not [.r,s.] c= ].p,q.] )
assume that
A1: q < s and
A2: r <= s ; ::_thesis: not [.r,s.] c= ].p,q.]
take t = s; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in ].p,q.] )
thus t in [.r,s.] by A2, Th1; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A1, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:122
for q, s, r, p being ext-real number st q < s & r < s holds
not [.r,s.[ c= ].p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not [.r,s.[ c= ].p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not [.r,s.[ c= ].p,q.]
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not [.r,s.[ c= ].p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in ].p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in [.r,s.[ by A5, Th3; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A4, Th2; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not [.r,s.[ c= ].p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in ].p,q.] )
thus t in [.r,s.[ by A7, A8, Th3; ::_thesis: not t in ].p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:123
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.] c= ].p,q.]
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.] c= ].p,q.] )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.] c= ].p,q.]
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.] c= ].p,q.]
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.] )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.] by A5, Th2; ::_thesis: not t in ].p,q.]
thus not t in ].p,q.] by A4, Th2; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not ].r,s.] c= ].p,q.]
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.] )
thus t in ].r,s.] by A7, A8, Th2; ::_thesis: not t in ].p,q.]
q < t by A6, A7, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:124
for q, s, r, p being ext-real number st q <= s & r <= s holds
not [.r,s.] c= ].p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q <= s & r <= s implies not [.r,s.] c= ].p,q.[ )
assume that
A1: q <= s and
A2: r <= s ; ::_thesis: not [.r,s.] c= ].p,q.[
take t = s; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.] & not t in ].p,q.[ )
thus t in [.r,s.] by A2, Th1; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A1, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:125
for q, s, r, p being ext-real number st q < s & r < s holds
not [.r,s.[ c= ].p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not [.r,s.[ c= ].p,q.[ )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not [.r,s.[ c= ].p,q.[
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not [.r,s.[ c= ].p,q.[
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in ].p,q.[ )
r < t by A3, A4, XXREAL_0:2;
hence t in [.r,s.[ by A5, Th3; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A4, Th4; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not [.r,s.[ c= ].p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in [.r,s.[ & not t in ].p,q.[ )
thus t in [.r,s.[ by A7, A8, Th3; ::_thesis: not t in ].p,q.[
q < t by A6, A7, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:126
for q, s, r, p being ext-real number st q <= s & r < s holds
not ].r,s.] c= ].p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q <= s & r < s implies not ].r,s.] c= ].p,q.[ )
assume that
A1: q <= s and
A2: r < s ; ::_thesis: not ].r,s.] c= ].p,q.[
take t = s; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.] & not t in ].p,q.[ )
thus t in ].r,s.] by A2, Th2; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A1, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:127
for q, s, r, p being ext-real number st q < s & r < s holds
not ].r,s.[ c= ].p,q.[
proof
let q, s, r, p be ext-real number ; ::_thesis: ( q < s & r < s implies not ].r,s.[ c= ].p,q.[ )
assume that
A1: q < s and
A2: r < s ; ::_thesis: not ].r,s.[ c= ].p,q.[
percases ( r <= q or q < r ) ;
supposeA3: r <= q ; ::_thesis: not ].r,s.[ c= ].p,q.[
consider t being ext-real number such that
A4: q < t and
A5: t < s by A1, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.[ )
r < t by A3, A4, XXREAL_0:2;
hence t in ].r,s.[ by A5, Th4; ::_thesis: not t in ].p,q.[
thus not t in ].p,q.[ by A4, Th4; ::_thesis: verum
end;
supposeA6: q < r ; ::_thesis: not ].r,s.[ c= ].p,q.[
consider t being ext-real number such that
A7: r < t and
A8: t < s by A2, XREAL_1:227;
take t ; :: according to MEMBERED:def_8 ::_thesis: ( t in ].r,s.[ & not t in ].p,q.[ )
thus t in ].r,s.[ by A7, A8, Th4; ::_thesis: not t in ].p,q.[
q < t by A6, A7, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
end;
end;
begin
theorem Th128: :: XXREAL_1:128
for r, s being ext-real number st r <= s holds
[.r,s.] = ].r,s.[ \/ {r,s}
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] = ].r,s.[ \/ {r,s} )
assume A1: r <= s ; ::_thesis: [.r,s.] = ].r,s.[ \/ {r,s}
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.] or t in ].r,s.[ \/ {r,s} ) & ( not t in ].r,s.[ \/ {r,s} or t in [.r,s.] ) )
thus ( t in [.r,s.] implies t in ].r,s.[ \/ {r,s} ) ::_thesis: ( not t in ].r,s.[ \/ {r,s} or t in [.r,s.] )
proof
assume t in [.r,s.] ; ::_thesis: t in ].r,s.[ \/ {r,s}
then ( t in ].r,s.[ or t = r or t = s ) by Th5;
then ( t in ].r,s.[ or t in {r,s} ) by TARSKI:def_2;
hence t in ].r,s.[ \/ {r,s} by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in ].r,s.[ \/ {r,s} ; ::_thesis: t in [.r,s.]
then ( t in ].r,s.[ or t in {r,s} ) by XBOOLE_0:def_3;
then ( t in ].r,s.[ or t = r or t = s ) by TARSKI:def_2;
hence t in [.r,s.] by A1, Th1, Th16; ::_thesis: verum
end;
theorem Th129: :: XXREAL_1:129
for r, s being ext-real number st r <= s holds
[.r,s.] = [.r,s.[ \/ {s}
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] = [.r,s.[ \/ {s} )
assume A1: r <= s ; ::_thesis: [.r,s.] = [.r,s.[ \/ {s}
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.] or t in [.r,s.[ \/ {s} ) & ( not t in [.r,s.[ \/ {s} or t in [.r,s.] ) )
thus ( t in [.r,s.] implies t in [.r,s.[ \/ {s} ) ::_thesis: ( not t in [.r,s.[ \/ {s} or t in [.r,s.] )
proof
assume t in [.r,s.] ; ::_thesis: t in [.r,s.[ \/ {s}
then ( t in [.r,s.[ or t = s ) by Th7;
then ( t in [.r,s.[ or t in {s} ) by TARSKI:def_1;
hence t in [.r,s.[ \/ {s} by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in [.r,s.[ \/ {s} ; ::_thesis: t in [.r,s.]
then ( t in [.r,s.[ or t in {s} ) by XBOOLE_0:def_3;
then ( t in [.r,s.[ or t = s ) by TARSKI:def_1;
hence t in [.r,s.] by A1, Th1, Th13; ::_thesis: verum
end;
theorem Th130: :: XXREAL_1:130
for r, s being ext-real number st r <= s holds
[.r,s.] = {r} \/ ].r,s.]
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] = {r} \/ ].r,s.] )
assume A1: r <= s ; ::_thesis: [.r,s.] = {r} \/ ].r,s.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.] or t in {r} \/ ].r,s.] ) & ( not t in {r} \/ ].r,s.] or t in [.r,s.] ) )
thus ( t in [.r,s.] implies t in {r} \/ ].r,s.] ) ::_thesis: ( not t in {r} \/ ].r,s.] or t in [.r,s.] )
proof
assume t in [.r,s.] ; ::_thesis: t in {r} \/ ].r,s.]
then ( t in ].r,s.] or t = r ) by Th6;
then ( t in ].r,s.] or t in {r} ) by TARSKI:def_1;
hence t in {r} \/ ].r,s.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in {r} \/ ].r,s.] ; ::_thesis: t in [.r,s.]
then ( t in ].r,s.] or t in {r} ) by XBOOLE_0:def_3;
then ( t in ].r,s.] or t = r ) by TARSKI:def_1;
hence t in [.r,s.] by A1, Th1, Th12; ::_thesis: verum
end;
theorem Th131: :: XXREAL_1:131
for r, s being ext-real number st r < s holds
[.r,s.[ = {r} \/ ].r,s.[
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies [.r,s.[ = {r} \/ ].r,s.[ )
assume A1: r < s ; ::_thesis: [.r,s.[ = {r} \/ ].r,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ or t in {r} \/ ].r,s.[ ) & ( not t in {r} \/ ].r,s.[ or t in [.r,s.[ ) )
thus ( t in [.r,s.[ implies t in {r} \/ ].r,s.[ ) ::_thesis: ( not t in {r} \/ ].r,s.[ or t in [.r,s.[ )
proof
assume t in [.r,s.[ ; ::_thesis: t in {r} \/ ].r,s.[
then ( t in ].r,s.[ or t = r ) by Th8;
then ( t in ].r,s.[ or t in {r} ) by TARSKI:def_1;
hence t in {r} \/ ].r,s.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in {r} \/ ].r,s.[ ; ::_thesis: t in [.r,s.[
then ( t in ].r,s.[ or t in {r} ) by XBOOLE_0:def_3;
then ( t in ].r,s.[ or t = r ) by TARSKI:def_1;
hence t in [.r,s.[ by A1, Th3, Th14; ::_thesis: verum
end;
theorem Th132: :: XXREAL_1:132
for r, s being ext-real number st r < s holds
].r,s.] = ].r,s.[ \/ {s}
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies ].r,s.] = ].r,s.[ \/ {s} )
assume A1: r < s ; ::_thesis: ].r,s.] = ].r,s.[ \/ {s}
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] or t in ].r,s.[ \/ {s} ) & ( not t in ].r,s.[ \/ {s} or t in ].r,s.] ) )
thus ( t in ].r,s.] implies t in ].r,s.[ \/ {s} ) ::_thesis: ( not t in ].r,s.[ \/ {s} or t in ].r,s.] )
proof
assume t in ].r,s.] ; ::_thesis: t in ].r,s.[ \/ {s}
then ( t in ].r,s.[ or t = s ) by Th9;
then ( t in ].r,s.[ or t in {s} ) by TARSKI:def_1;
hence t in ].r,s.[ \/ {s} by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in ].r,s.[ \/ {s} ; ::_thesis: t in ].r,s.]
then ( t in ].r,s.[ or t in {s} ) by XBOOLE_0:def_3;
then ( t in ].r,s.[ or t = s ) by TARSKI:def_1;
hence t in ].r,s.] by A1, Th2, Th15; ::_thesis: verum
end;
theorem :: XXREAL_1:133
for r, s being ext-real number st r <= s holds
[.r,s.] \ {r,s} = ].r,s.[
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] \ {r,s} = ].r,s.[ )
assume r <= s ; ::_thesis: [.r,s.] \ {r,s} = ].r,s.[
then A1: [.r,s.] = ].r,s.[ \/ {r,s} by Th128;
A2: not r in ].r,s.[ by Th4;
not s in ].r,s.[ by Th4;
hence [.r,s.] \ {r,s} = ].r,s.[ by A1, A2, ZFMISC_1:121; ::_thesis: verum
end;
theorem :: XXREAL_1:134
for r, s being ext-real number st r <= s holds
[.r,s.] \ {r} = ].r,s.]
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] \ {r} = ].r,s.] )
assume r <= s ; ::_thesis: [.r,s.] \ {r} = ].r,s.]
then A1: [.r,s.] = {r} \/ ].r,s.] by Th130;
not r in ].r,s.] by Th2;
hence [.r,s.] \ {r} = ].r,s.] by A1, ZFMISC_1:117; ::_thesis: verum
end;
theorem :: XXREAL_1:135
for r, s being ext-real number st r <= s holds
[.r,s.] \ {s} = [.r,s.[
proof
let r, s be ext-real number ; ::_thesis: ( r <= s implies [.r,s.] \ {s} = [.r,s.[ )
assume r <= s ; ::_thesis: [.r,s.] \ {s} = [.r,s.[
then A1: [.r,s.] = [.r,s.[ \/ {s} by Th129;
not s in [.r,s.[ by Th3;
hence [.r,s.] \ {s} = [.r,s.[ by A1, ZFMISC_1:117; ::_thesis: verum
end;
theorem :: XXREAL_1:136
for r, s being ext-real number st r < s holds
[.r,s.[ \ {r} = ].r,s.[
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies [.r,s.[ \ {r} = ].r,s.[ )
assume r < s ; ::_thesis: [.r,s.[ \ {r} = ].r,s.[
then A1: [.r,s.[ = {r} \/ ].r,s.[ by Th131;
not r in ].r,s.[ by Th4;
hence [.r,s.[ \ {r} = ].r,s.[ by A1, ZFMISC_1:117; ::_thesis: verum
end;
theorem :: XXREAL_1:137
for r, s being ext-real number st r < s holds
].r,s.] \ {s} = ].r,s.[
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies ].r,s.] \ {s} = ].r,s.[ )
assume r < s ; ::_thesis: ].r,s.] \ {s} = ].r,s.[
then A1: ].r,s.] = ].r,s.[ \/ {s} by Th132;
not s in ].r,s.[ by Th4;
hence ].r,s.] \ {s} = ].r,s.[ by A1, ZFMISC_1:117; ::_thesis: verum
end;
theorem :: XXREAL_1:138
for r, s, t being ext-real number st r < s & s < t holds
].r,s.] /\ [.s,t.[ = {s}
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s < t implies ].r,s.] /\ [.s,t.[ = {s} )
assume that
A1: r < s and
A2: s < t ; ::_thesis: ].r,s.] /\ [.s,t.[ = {s}
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_].r,s.]_/\_[.s,t.[_implies_x_=_s_)_&_(_x_=_s_implies_x_in_].r,s.]_/\_[.s,t.[_)_)
let x be set ; ::_thesis: ( ( x in ].r,s.] /\ [.s,t.[ implies x = s ) & ( x = s implies x in ].r,s.] /\ [.s,t.[ ) )
hereby ::_thesis: ( x = s implies x in ].r,s.] /\ [.s,t.[ )
assume A3: x in ].r,s.] /\ [.s,t.[ ; ::_thesis: x = s
then reconsider p = x as ext-real number ;
A4: p in ].r,s.] by A3, XBOOLE_0:def_4;
p in [.s,t.[ by A3, XBOOLE_0:def_4;
then A5: s <= p by Th3;
p <= s by A4, Th2;
hence x = s by A5, XXREAL_0:1; ::_thesis: verum
end;
assume A6: x = s ; ::_thesis: x in ].r,s.] /\ [.s,t.[
A7: s in ].r,s.] by A1, Th2;
s in [.s,t.[ by A2, Th3;
hence x in ].r,s.] /\ [.s,t.[ by A6, A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence ].r,s.] /\ [.s,t.[ = {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:139
for r, s, p, q being ext-real number holds [.r,s.[ /\ [.p,q.[ = [.(max (r,p)),(min (s,q)).[
proof
let r, s, p, q be ext-real number ; ::_thesis: [.r,s.[ /\ [.p,q.[ = [.(max (r,p)),(min (s,q)).[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ /\ [.p,q.[ or t in [.(max (r,p)),(min (s,q)).[ ) & ( not t in [.(max (r,p)),(min (s,q)).[ or t in [.r,s.[ /\ [.p,q.[ ) )
thus ( t in [.r,s.[ /\ [.p,q.[ implies t in [.(max (r,p)),(min (s,q)).[ ) ::_thesis: ( not t in [.(max (r,p)),(min (s,q)).[ or t in [.r,s.[ /\ [.p,q.[ )
proof
assume A1: t in [.r,s.[ /\ [.p,q.[ ; ::_thesis: t in [.(max (r,p)),(min (s,q)).[
then A2: t in [.r,s.[ by XBOOLE_0:def_4;
A3: t in [.p,q.[ by A1, XBOOLE_0:def_4;
A4: r <= t by A2, Th3;
A5: t < s by A2, Th3;
A6: p <= t by A3, Th3;
A7: t < q by A3, Th3;
A8: max (r,p) <= t by A4, A6, XXREAL_0:28;
t < min (s,q) by A5, A7, XXREAL_0:21;
hence t in [.(max (r,p)),(min (s,q)).[ by A8, Th3; ::_thesis: verum
end;
assume A9: t in [.(max (r,p)),(min (s,q)).[ ; ::_thesis: t in [.r,s.[ /\ [.p,q.[
then A10: max (r,p) <= t by Th3;
A11: t < min (s,q) by A9, Th3;
A12: r <= t by A10, XXREAL_0:30;
A13: p <= t by A10, XXREAL_0:30;
A14: t < s by A11, XXREAL_0:23;
A15: t < q by A11, XXREAL_0:23;
A16: t in [.r,s.[ by A12, A14, Th3;
t in [.p,q.[ by A13, A15, Th3;
hence t in [.r,s.[ /\ [.p,q.[ by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:140
for r, s, p, q being ext-real number holds [.r,s.] /\ [.p,q.] = [.(max (r,p)),(min (s,q)).]
proof
let r, s, p, q be ext-real number ; ::_thesis: [.r,s.] /\ [.p,q.] = [.(max (r,p)),(min (s,q)).]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.] /\ [.p,q.] or t in [.(max (r,p)),(min (s,q)).] ) & ( not t in [.(max (r,p)),(min (s,q)).] or t in [.r,s.] /\ [.p,q.] ) )
thus ( t in [.r,s.] /\ [.p,q.] implies t in [.(max (r,p)),(min (s,q)).] ) ::_thesis: ( not t in [.(max (r,p)),(min (s,q)).] or t in [.r,s.] /\ [.p,q.] )
proof
assume A1: t in [.r,s.] /\ [.p,q.] ; ::_thesis: t in [.(max (r,p)),(min (s,q)).]
then A2: t in [.r,s.] by XBOOLE_0:def_4;
A3: t in [.p,q.] by A1, XBOOLE_0:def_4;
A4: r <= t by A2, Th1;
A5: t <= s by A2, Th1;
A6: p <= t by A3, Th1;
A7: t <= q by A3, Th1;
A8: max (r,p) <= t by A4, A6, XXREAL_0:28;
t <= min (s,q) by A5, A7, XXREAL_0:20;
hence t in [.(max (r,p)),(min (s,q)).] by A8, Th1; ::_thesis: verum
end;
assume A9: t in [.(max (r,p)),(min (s,q)).] ; ::_thesis: t in [.r,s.] /\ [.p,q.]
then A10: max (r,p) <= t by Th1;
A11: t <= min (s,q) by A9, Th1;
A12: r <= t by A10, XXREAL_0:30;
A13: p <= t by A10, XXREAL_0:30;
A14: t <= s by A11, XXREAL_0:22;
A15: t <= q by A11, XXREAL_0:22;
A16: t in [.r,s.] by A12, A14, Th1;
t in [.p,q.] by A13, A15, Th1;
hence t in [.r,s.] /\ [.p,q.] by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:141
for r, s, p, q being ext-real number holds ].r,s.] /\ ].p,q.] = ].(max (r,p)),(min (s,q)).]
proof
let r, s, p, q be ext-real number ; ::_thesis: ].r,s.] /\ ].p,q.] = ].(max (r,p)),(min (s,q)).]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ ].p,q.] or t in ].(max (r,p)),(min (s,q)).] ) & ( not t in ].(max (r,p)),(min (s,q)).] or t in ].r,s.] /\ ].p,q.] ) )
thus ( t in ].r,s.] /\ ].p,q.] implies t in ].(max (r,p)),(min (s,q)).] ) ::_thesis: ( not t in ].(max (r,p)),(min (s,q)).] or t in ].r,s.] /\ ].p,q.] )
proof
assume A1: t in ].r,s.] /\ ].p,q.] ; ::_thesis: t in ].(max (r,p)),(min (s,q)).]
then A2: t in ].r,s.] by XBOOLE_0:def_4;
A3: t in ].p,q.] by A1, XBOOLE_0:def_4;
A4: r < t by A2, Th2;
A5: t <= s by A2, Th2;
A6: p < t by A3, Th2;
A7: t <= q by A3, Th2;
A8: max (r,p) < t by A4, A6, XXREAL_0:29;
t <= min (s,q) by A5, A7, XXREAL_0:20;
hence t in ].(max (r,p)),(min (s,q)).] by A8, Th2; ::_thesis: verum
end;
assume A9: t in ].(max (r,p)),(min (s,q)).] ; ::_thesis: t in ].r,s.] /\ ].p,q.]
then A10: max (r,p) < t by Th2;
A11: t <= min (s,q) by A9, Th2;
A12: r < t by A10, XXREAL_0:31;
A13: p < t by A10, XXREAL_0:31;
A14: t <= s by A11, XXREAL_0:22;
A15: t <= q by A11, XXREAL_0:22;
A16: t in ].r,s.] by A12, A14, Th2;
t in ].p,q.] by A13, A15, Th2;
hence t in ].r,s.] /\ ].p,q.] by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:142
for r, s, p, q being ext-real number holds ].r,s.[ /\ ].p,q.[ = ].(max (r,p)),(min (s,q)).[
proof
let r, s, p, q be ext-real number ; ::_thesis: ].r,s.[ /\ ].p,q.[ = ].(max (r,p)),(min (s,q)).[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ ].p,q.[ or t in ].(max (r,p)),(min (s,q)).[ ) & ( not t in ].(max (r,p)),(min (s,q)).[ or t in ].r,s.[ /\ ].p,q.[ ) )
thus ( t in ].r,s.[ /\ ].p,q.[ implies t in ].(max (r,p)),(min (s,q)).[ ) ::_thesis: ( not t in ].(max (r,p)),(min (s,q)).[ or t in ].r,s.[ /\ ].p,q.[ )
proof
assume A1: t in ].r,s.[ /\ ].p,q.[ ; ::_thesis: t in ].(max (r,p)),(min (s,q)).[
then A2: t in ].r,s.[ by XBOOLE_0:def_4;
A3: t in ].p,q.[ by A1, XBOOLE_0:def_4;
A4: r < t by A2, Th4;
A5: t < s by A2, Th4;
A6: p < t by A3, Th4;
A7: t < q by A3, Th4;
A8: max (r,p) < t by A4, A6, XXREAL_0:29;
t < min (s,q) by A5, A7, XXREAL_0:21;
hence t in ].(max (r,p)),(min (s,q)).[ by A8, Th4; ::_thesis: verum
end;
assume A9: t in ].(max (r,p)),(min (s,q)).[ ; ::_thesis: t in ].r,s.[ /\ ].p,q.[
then A10: max (r,p) < t by Th4;
A11: t < min (s,q) by A9, Th4;
A12: r < t by A10, XXREAL_0:31;
A13: p < t by A10, XXREAL_0:31;
A14: t < s by A11, XXREAL_0:23;
A15: t < q by A11, XXREAL_0:23;
A16: t in ].r,s.[ by A12, A14, Th4;
t in ].p,q.[ by A13, A15, Th4;
hence t in ].r,s.[ /\ ].p,q.[ by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:143
for r, p, s, q being ext-real number st r <= p & s <= q holds
[.r,s.] /\ [.p,q.] = [.p,s.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies [.r,s.] /\ [.p,q.] = [.p,s.] )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: [.r,s.] /\ [.p,q.] = [.p,s.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.] /\ [.p,q.] or t in [.p,s.] ) & ( not t in [.p,s.] or t in [.r,s.] /\ [.p,q.] ) )
thus ( t in [.r,s.] /\ [.p,q.] implies t in [.p,s.] ) ::_thesis: ( not t in [.p,s.] or t in [.r,s.] /\ [.p,q.] )
proof
assume A3: t in [.r,s.] /\ [.p,q.] ; ::_thesis: t in [.p,s.]
then A4: t in [.r,s.] by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: t <= s by A4, Th1;
p <= t by A5, Th1;
hence t in [.p,s.] by A6, Th1; ::_thesis: verum
end;
assume A7: t in [.p,s.] ; ::_thesis: t in [.r,s.] /\ [.p,q.]
then A8: p <= t by Th1;
A9: t <= s by A7, Th1;
A10: r <= t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in [.r,s.] by A9, A10, Th1;
t in [.p,q.] by A8, A11, Th1;
hence t in [.r,s.] /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:144
for r, p, s, q being ext-real number st r <= p & s <= q holds
[.r,s.[ /\ [.p,q.] = [.p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies [.r,s.[ /\ [.p,q.] = [.p,s.[ )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: [.r,s.[ /\ [.p,q.] = [.p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ /\ [.p,q.] or t in [.p,s.[ ) & ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.] ) )
thus ( t in [.r,s.[ /\ [.p,q.] implies t in [.p,s.[ ) ::_thesis: ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.] )
proof
assume A3: t in [.r,s.[ /\ [.p,q.] ; ::_thesis: t in [.p,s.[
then A4: t in [.r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th3;
p <= t by A5, Th1;
hence t in [.p,s.[ by A6, Th3; ::_thesis: verum
end;
assume A7: t in [.p,s.[ ; ::_thesis: t in [.r,s.[ /\ [.p,q.]
then A8: p <= t by Th3;
A9: t < s by A7, Th3;
A10: r <= t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in [.r,s.[ by A9, A10, Th3;
t in [.p,q.] by A8, A11, Th1;
hence t in [.r,s.[ /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:145
for r, p, s, q being ext-real number st r >= p & s > q holds
[.r,s.[ /\ [.p,q.] = [.r,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s > q implies [.r,s.[ /\ [.p,q.] = [.r,q.] )
assume that
A1: r >= p and
A2: s > q ; ::_thesis: [.r,s.[ /\ [.p,q.] = [.r,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ /\ [.p,q.] or t in [.r,q.] ) & ( not t in [.r,q.] or t in [.r,s.[ /\ [.p,q.] ) )
thus ( t in [.r,s.[ /\ [.p,q.] implies t in [.r,q.] ) ::_thesis: ( not t in [.r,q.] or t in [.r,s.[ /\ [.p,q.] )
proof
assume A3: t in [.r,s.[ /\ [.p,q.] ; ::_thesis: t in [.r,q.]
then A4: t in [.r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: r <= t by A4, Th3;
t <= q by A5, Th1;
hence t in [.r,q.] by A6, Th1; ::_thesis: verum
end;
assume A7: t in [.r,q.] ; ::_thesis: t in [.r,s.[ /\ [.p,q.]
then A8: r <= t by Th1;
A9: t <= q by A7, Th1;
then A10: t < s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in [.r,s.[ by A8, A10, Th3;
t in [.p,q.] by A9, A11, Th1;
hence t in [.r,s.[ /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:146
for r, p, s, q being ext-real number st r < p & s <= q holds
].r,s.] /\ [.p,q.] = [.p,s.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & s <= q implies ].r,s.] /\ [.p,q.] = [.p,s.] )
assume that
A1: r < p and
A2: s <= q ; ::_thesis: ].r,s.] /\ [.p,q.] = [.p,s.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ [.p,q.] or t in [.p,s.] ) & ( not t in [.p,s.] or t in ].r,s.] /\ [.p,q.] ) )
thus ( t in ].r,s.] /\ [.p,q.] implies t in [.p,s.] ) ::_thesis: ( not t in [.p,s.] or t in ].r,s.] /\ [.p,q.] )
proof
assume A3: t in ].r,s.] /\ [.p,q.] ; ::_thesis: t in [.p,s.]
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: t <= s by A4, Th2;
p <= t by A5, Th1;
hence t in [.p,s.] by A6, Th1; ::_thesis: verum
end;
assume A7: t in [.p,s.] ; ::_thesis: t in ].r,s.] /\ [.p,q.]
then A8: p <= t by Th1;
A9: t <= s by A7, Th1;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.] by A9, A10, Th2;
t in [.p,q.] by A8, A11, Th1;
hence t in ].r,s.] /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:147
for r, p, s, q being ext-real number st r >= p & s >= q holds
].r,s.] /\ [.p,q.] = ].r,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s >= q implies ].r,s.] /\ [.p,q.] = ].r,q.] )
assume that
A1: r >= p and
A2: s >= q ; ::_thesis: ].r,s.] /\ [.p,q.] = ].r,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ [.p,q.] or t in ].r,q.] ) & ( not t in ].r,q.] or t in ].r,s.] /\ [.p,q.] ) )
thus ( t in ].r,s.] /\ [.p,q.] implies t in ].r,q.] ) ::_thesis: ( not t in ].r,q.] or t in ].r,s.] /\ [.p,q.] )
proof
assume A3: t in ].r,s.] /\ [.p,q.] ; ::_thesis: t in ].r,q.]
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th2;
t <= q by A5, Th1;
hence t in ].r,q.] by A6, Th2; ::_thesis: verum
end;
assume A7: t in ].r,q.] ; ::_thesis: t in ].r,s.] /\ [.p,q.]
then A8: r < t by Th2;
A9: t <= q by A7, Th2;
then A10: t <= s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.] by A8, A10, Th2;
t in [.p,q.] by A9, A11, Th1;
hence t in ].r,s.] /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:148
for r, p, s, q being ext-real number st r < p & s <= q holds
].r,s.[ /\ [.p,q.] = [.p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & s <= q implies ].r,s.[ /\ [.p,q.] = [.p,s.[ )
assume that
A1: r < p and
A2: s <= q ; ::_thesis: ].r,s.[ /\ [.p,q.] = [.p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ [.p,q.] or t in [.p,s.[ ) & ( not t in [.p,s.[ or t in ].r,s.[ /\ [.p,q.] ) )
thus ( t in ].r,s.[ /\ [.p,q.] implies t in [.p,s.[ ) ::_thesis: ( not t in [.p,s.[ or t in ].r,s.[ /\ [.p,q.] )
proof
assume A3: t in ].r,s.[ /\ [.p,q.] ; ::_thesis: t in [.p,s.[
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th4;
p <= t by A5, Th1;
hence t in [.p,s.[ by A6, Th3; ::_thesis: verum
end;
assume A7: t in [.p,s.[ ; ::_thesis: t in ].r,s.[ /\ [.p,q.]
then A8: p <= t by Th3;
A9: t < s by A7, Th3;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.[ by A9, A10, Th4;
t in [.p,q.] by A8, A11, Th1;
hence t in ].r,s.[ /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:149
for r, p, s, q being ext-real number st r >= p & s > q holds
].r,s.[ /\ [.p,q.] = ].r,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s > q implies ].r,s.[ /\ [.p,q.] = ].r,q.] )
assume that
A1: r >= p and
A2: s > q ; ::_thesis: ].r,s.[ /\ [.p,q.] = ].r,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ [.p,q.] or t in ].r,q.] ) & ( not t in ].r,q.] or t in ].r,s.[ /\ [.p,q.] ) )
thus ( t in ].r,s.[ /\ [.p,q.] implies t in ].r,q.] ) ::_thesis: ( not t in ].r,q.] or t in ].r,s.[ /\ [.p,q.] )
proof
assume A3: t in ].r,s.[ /\ [.p,q.] ; ::_thesis: t in ].r,q.]
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.] by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th4;
t <= q by A5, Th1;
hence t in ].r,q.] by A6, Th2; ::_thesis: verum
end;
assume A7: t in ].r,q.] ; ::_thesis: t in ].r,s.[ /\ [.p,q.]
then A8: r < t by Th2;
A9: t <= q by A7, Th2;
then A10: t < s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.[ by A8, A10, Th4;
t in [.p,q.] by A9, A11, Th1;
hence t in ].r,s.[ /\ [.p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:150
for r, p, s, q being ext-real number st r <= p & s <= q holds
[.r,s.[ /\ [.p,q.[ = [.p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies [.r,s.[ /\ [.p,q.[ = [.p,s.[ )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: [.r,s.[ /\ [.p,q.[ = [.p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ /\ [.p,q.[ or t in [.p,s.[ ) & ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.[ ) )
thus ( t in [.r,s.[ /\ [.p,q.[ implies t in [.p,s.[ ) ::_thesis: ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.[ )
proof
assume A3: t in [.r,s.[ /\ [.p,q.[ ; ::_thesis: t in [.p,s.[
then A4: t in [.r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th3;
p <= t by A5, Th3;
hence t in [.p,s.[ by A6, Th3; ::_thesis: verum
end;
assume A7: t in [.p,s.[ ; ::_thesis: t in [.r,s.[ /\ [.p,q.[
then A8: p <= t by Th3;
A9: t < s by A7, Th3;
A10: r <= t by A1, A8, XXREAL_0:2;
A11: t < q by A2, A9, XXREAL_0:2;
A12: t in [.r,s.[ by A9, A10, Th3;
t in [.p,q.[ by A8, A11, Th3;
hence t in [.r,s.[ /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:151
for r, p, s, q being ext-real number st r >= p & s >= q holds
[.r,s.[ /\ [.p,q.[ = [.r,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s >= q implies [.r,s.[ /\ [.p,q.[ = [.r,q.[ )
assume that
A1: r >= p and
A2: s >= q ; ::_thesis: [.r,s.[ /\ [.p,q.[ = [.r,q.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ /\ [.p,q.[ or t in [.r,q.[ ) & ( not t in [.r,q.[ or t in [.r,s.[ /\ [.p,q.[ ) )
thus ( t in [.r,s.[ /\ [.p,q.[ implies t in [.r,q.[ ) ::_thesis: ( not t in [.r,q.[ or t in [.r,s.[ /\ [.p,q.[ )
proof
assume A3: t in [.r,s.[ /\ [.p,q.[ ; ::_thesis: t in [.r,q.[
then A4: t in [.r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: r <= t by A4, Th3;
t < q by A5, Th3;
hence t in [.r,q.[ by A6, Th3; ::_thesis: verum
end;
assume A7: t in [.r,q.[ ; ::_thesis: t in [.r,s.[ /\ [.p,q.[
then A8: r <= t by Th3;
A9: t < q by A7, Th3;
then A10: t < s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in [.r,s.[ by A8, A10, Th3;
t in [.p,q.[ by A9, A11, Th3;
hence t in [.r,s.[ /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th152: :: XXREAL_1:152
for r, p, s, q being ext-real number st r < p & s < q holds
].r,s.] /\ [.p,q.[ = [.p,s.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & s < q implies ].r,s.] /\ [.p,q.[ = [.p,s.] )
assume that
A1: r < p and
A2: s < q ; ::_thesis: ].r,s.] /\ [.p,q.[ = [.p,s.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ [.p,q.[ or t in [.p,s.] ) & ( not t in [.p,s.] or t in ].r,s.] /\ [.p,q.[ ) )
thus ( t in ].r,s.] /\ [.p,q.[ implies t in [.p,s.] ) ::_thesis: ( not t in [.p,s.] or t in ].r,s.] /\ [.p,q.[ )
proof
assume A3: t in ].r,s.] /\ [.p,q.[ ; ::_thesis: t in [.p,s.]
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: t <= s by A4, Th2;
p <= t by A5, Th3;
hence t in [.p,s.] by A6, Th1; ::_thesis: verum
end;
assume A7: t in [.p,s.] ; ::_thesis: t in ].r,s.] /\ [.p,q.[
then A8: p <= t by Th1;
A9: t <= s by A7, Th1;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t < q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.] by A9, A10, Th2;
t in [.p,q.[ by A8, A11, Th3;
hence t in ].r,s.] /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:153
for r, p, s, q being ext-real number st r >= p & s >= q holds
].r,s.] /\ [.p,q.[ = ].r,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s >= q implies ].r,s.] /\ [.p,q.[ = ].r,q.[ )
assume that
A1: r >= p and
A2: s >= q ; ::_thesis: ].r,s.] /\ [.p,q.[ = ].r,q.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ [.p,q.[ or t in ].r,q.[ ) & ( not t in ].r,q.[ or t in ].r,s.] /\ [.p,q.[ ) )
thus ( t in ].r,s.] /\ [.p,q.[ implies t in ].r,q.[ ) ::_thesis: ( not t in ].r,q.[ or t in ].r,s.] /\ [.p,q.[ )
proof
assume A3: t in ].r,s.] /\ [.p,q.[ ; ::_thesis: t in ].r,q.[
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th2;
t < q by A5, Th3;
hence t in ].r,q.[ by A6, Th4; ::_thesis: verum
end;
assume A7: t in ].r,q.[ ; ::_thesis: t in ].r,s.] /\ [.p,q.[
then A8: r < t by Th4;
A9: t < q by A7, Th4;
then A10: t <= s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.] by A8, A10, Th2;
t in [.p,q.[ by A9, A11, Th3;
hence t in ].r,s.] /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th154: :: XXREAL_1:154
for r, p, s, q being ext-real number st r < p & s <= q holds
].r,s.[ /\ [.p,q.[ = [.p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r < p & s <= q implies ].r,s.[ /\ [.p,q.[ = [.p,s.[ )
assume that
A1: r < p and
A2: s <= q ; ::_thesis: ].r,s.[ /\ [.p,q.[ = [.p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ [.p,q.[ or t in [.p,s.[ ) & ( not t in [.p,s.[ or t in ].r,s.[ /\ [.p,q.[ ) )
thus ( t in ].r,s.[ /\ [.p,q.[ implies t in [.p,s.[ ) ::_thesis: ( not t in [.p,s.[ or t in ].r,s.[ /\ [.p,q.[ )
proof
assume A3: t in ].r,s.[ /\ [.p,q.[ ; ::_thesis: t in [.p,s.[
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th4;
p <= t by A5, Th3;
hence t in [.p,s.[ by A6, Th3; ::_thesis: verum
end;
assume A7: t in [.p,s.[ ; ::_thesis: t in ].r,s.[ /\ [.p,q.[
then A8: p <= t by Th3;
A9: t < s by A7, Th3;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t < q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.[ by A9, A10, Th4;
t in [.p,q.[ by A8, A11, Th3;
hence t in ].r,s.[ /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:155
for r, p, s, q being ext-real number st r >= p & s >= q holds
].r,s.[ /\ [.p,q.[ = ].r,q.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s >= q implies ].r,s.[ /\ [.p,q.[ = ].r,q.[ )
assume that
A1: r >= p and
A2: s >= q ; ::_thesis: ].r,s.[ /\ [.p,q.[ = ].r,q.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ [.p,q.[ or t in ].r,q.[ ) & ( not t in ].r,q.[ or t in ].r,s.[ /\ [.p,q.[ ) )
thus ( t in ].r,s.[ /\ [.p,q.[ implies t in ].r,q.[ ) ::_thesis: ( not t in ].r,q.[ or t in ].r,s.[ /\ [.p,q.[ )
proof
assume A3: t in ].r,s.[ /\ [.p,q.[ ; ::_thesis: t in ].r,q.[
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th4;
t < q by A5, Th3;
hence t in ].r,q.[ by A6, Th4; ::_thesis: verum
end;
assume A7: t in ].r,q.[ ; ::_thesis: t in ].r,s.[ /\ [.p,q.[
then A8: r < t by Th4;
A9: t < q by A7, Th4;
then A10: t < s by A2, XXREAL_0:2;
A11: p <= t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.[ by A8, A10, Th4;
t in [.p,q.[ by A9, A11, Th3;
hence t in ].r,s.[ /\ [.p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:156
for r, p, s, q being ext-real number st r <= p & s <= q holds
].r,s.] /\ ].p,q.] = ].p,s.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies ].r,s.] /\ ].p,q.] = ].p,s.] )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: ].r,s.] /\ ].p,q.] = ].p,s.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ ].p,q.] or t in ].p,s.] ) & ( not t in ].p,s.] or t in ].r,s.] /\ ].p,q.] ) )
thus ( t in ].r,s.] /\ ].p,q.] implies t in ].p,s.] ) ::_thesis: ( not t in ].p,s.] or t in ].r,s.] /\ ].p,q.] )
proof
assume A3: t in ].r,s.] /\ ].p,q.] ; ::_thesis: t in ].p,s.]
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in ].p,q.] by A3, XBOOLE_0:def_4;
A6: t <= s by A4, Th2;
p < t by A5, Th2;
hence t in ].p,s.] by A6, Th2; ::_thesis: verum
end;
assume A7: t in ].p,s.] ; ::_thesis: t in ].r,s.] /\ ].p,q.]
then A8: p < t by Th2;
A9: t <= s by A7, Th2;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.] by A9, A10, Th2;
t in ].p,q.] by A8, A11, Th2;
hence t in ].r,s.] /\ ].p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:157
for r, p, s, q being ext-real number st r >= p & s >= q holds
].r,s.] /\ ].p,q.] = ].r,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s >= q implies ].r,s.] /\ ].p,q.] = ].r,q.] )
assume that
A1: r >= p and
A2: s >= q ; ::_thesis: ].r,s.] /\ ].p,q.] = ].r,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] /\ ].p,q.] or t in ].r,q.] ) & ( not t in ].r,q.] or t in ].r,s.] /\ ].p,q.] ) )
thus ( t in ].r,s.] /\ ].p,q.] implies t in ].r,q.] ) ::_thesis: ( not t in ].r,q.] or t in ].r,s.] /\ ].p,q.] )
proof
assume A3: t in ].r,s.] /\ ].p,q.] ; ::_thesis: t in ].r,q.]
then A4: t in ].r,s.] by XBOOLE_0:def_4;
A5: t in ].p,q.] by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th2;
t <= q by A5, Th2;
hence t in ].r,q.] by A6, Th2; ::_thesis: verum
end;
assume A7: t in ].r,q.] ; ::_thesis: t in ].r,s.] /\ ].p,q.]
then A8: r < t by Th2;
A9: t <= q by A7, Th2;
then A10: t <= s by A2, XXREAL_0:2;
A11: p < t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.] by A8, A10, Th2;
t in ].p,q.] by A9, A11, Th2;
hence t in ].r,s.] /\ ].p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:158
for r, p, s, q being ext-real number st r <= p & s <= q holds
].r,s.[ /\ ].p,q.] = ].p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies ].r,s.[ /\ ].p,q.] = ].p,s.[ )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: ].r,s.[ /\ ].p,q.] = ].p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ ].p,q.] or t in ].p,s.[ ) & ( not t in ].p,s.[ or t in ].r,s.[ /\ ].p,q.] ) )
thus ( t in ].r,s.[ /\ ].p,q.] implies t in ].p,s.[ ) ::_thesis: ( not t in ].p,s.[ or t in ].r,s.[ /\ ].p,q.] )
proof
assume A3: t in ].r,s.[ /\ ].p,q.] ; ::_thesis: t in ].p,s.[
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in ].p,q.] by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th4;
p < t by A5, Th2;
hence t in ].p,s.[ by A6, Th4; ::_thesis: verum
end;
assume A7: t in ].p,s.[ ; ::_thesis: t in ].r,s.[ /\ ].p,q.]
then A8: p < t by Th4;
A9: t < s by A7, Th4;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.[ by A9, A10, Th4;
t in ].p,q.] by A8, A11, Th2;
hence t in ].r,s.[ /\ ].p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th159: :: XXREAL_1:159
for r, p, s, q being ext-real number st r >= p & s > q holds
].r,s.[ /\ ].p,q.] = ].r,q.]
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r >= p & s > q implies ].r,s.[ /\ ].p,q.] = ].r,q.] )
assume that
A1: r >= p and
A2: s > q ; ::_thesis: ].r,s.[ /\ ].p,q.] = ].r,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ ].p,q.] or t in ].r,q.] ) & ( not t in ].r,q.] or t in ].r,s.[ /\ ].p,q.] ) )
thus ( t in ].r,s.[ /\ ].p,q.] implies t in ].r,q.] ) ::_thesis: ( not t in ].r,q.] or t in ].r,s.[ /\ ].p,q.] )
proof
assume A3: t in ].r,s.[ /\ ].p,q.] ; ::_thesis: t in ].r,q.]
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in ].p,q.] by A3, XBOOLE_0:def_4;
A6: r < t by A4, Th4;
t <= q by A5, Th2;
hence t in ].r,q.] by A6, Th2; ::_thesis: verum
end;
assume A7: t in ].r,q.] ; ::_thesis: t in ].r,s.[ /\ ].p,q.]
then A8: r < t by Th2;
A9: t <= q by A7, Th2;
then A10: t < s by A2, XXREAL_0:2;
A11: p < t by A1, A8, XXREAL_0:2;
A12: t in ].r,s.[ by A8, A10, Th4;
t in ].p,q.] by A9, A11, Th2;
hence t in ].r,s.[ /\ ].p,q.] by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th160: :: XXREAL_1:160
for r, p, s, q being ext-real number st r <= p & s <= q holds
].r,s.[ /\ ].p,q.[ = ].p,s.[
proof
let r, p, s, q be ext-real number ; ::_thesis: ( r <= p & s <= q implies ].r,s.[ /\ ].p,q.[ = ].p,s.[ )
assume that
A1: r <= p and
A2: s <= q ; ::_thesis: ].r,s.[ /\ ].p,q.[ = ].p,s.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.[ /\ ].p,q.[ or t in ].p,s.[ ) & ( not t in ].p,s.[ or t in ].r,s.[ /\ ].p,q.[ ) )
thus ( t in ].r,s.[ /\ ].p,q.[ implies t in ].p,s.[ ) ::_thesis: ( not t in ].p,s.[ or t in ].r,s.[ /\ ].p,q.[ )
proof
assume A3: t in ].r,s.[ /\ ].p,q.[ ; ::_thesis: t in ].p,s.[
then A4: t in ].r,s.[ by XBOOLE_0:def_4;
A5: t in ].p,q.[ by A3, XBOOLE_0:def_4;
A6: t < s by A4, Th4;
p < t by A5, Th4;
hence t in ].p,s.[ by A6, Th4; ::_thesis: verum
end;
assume A7: t in ].p,s.[ ; ::_thesis: t in ].r,s.[ /\ ].p,q.[
then A8: p < t by Th4;
A9: t < s by A7, Th4;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t < q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.[ by A9, A10, Th4;
t in ].p,q.[ by A8, A11, Th4;
hence t in ].r,s.[ /\ ].p,q.[ by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: XXREAL_1:161
for r, s, p, q being ext-real number holds [.r,s.[ \/ [.p,q.[ c= [.(min (r,p)),(max (s,q)).[
proof
let r, s, p, q be ext-real number ; ::_thesis: [.r,s.[ \/ [.p,q.[ c= [.(min (r,p)),(max (s,q)).[
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in [.r,s.[ \/ [.p,q.[ or t in [.(min (r,p)),(max (s,q)).[ )
assume t in [.r,s.[ \/ [.p,q.[ ; ::_thesis: t in [.(min (r,p)),(max (s,q)).[
then ( t in [.r,s.[ or t in [.p,q.[ ) by XBOOLE_0:def_3;
then A1: ( ( r <= t & t < s ) or ( p <= t & t < q ) ) by Th3;
then A2: min (r,p) <= t by XXREAL_0:23;
t < max (s,q) by A1, XXREAL_0:30;
hence t in [.(min (r,p)),(max (s,q)).[ by A2, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:162
for r, s, p, q being ext-real number st [.r,s.[ meets [.p,q.[ holds
[.r,s.[ \/ [.p,q.[ = [.(min (r,p)),(max (s,q)).[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( [.r,s.[ meets [.p,q.[ implies [.r,s.[ \/ [.p,q.[ = [.(min (r,p)),(max (s,q)).[ )
assume [.r,s.[ meets [.p,q.[ ; ::_thesis: [.r,s.[ \/ [.p,q.[ = [.(min (r,p)),(max (s,q)).[
then consider u being ext-real number such that
A1: u in [.r,s.[ and
A2: u in [.p,q.[ by MEMBERED:def_20;
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.r,s.[ \/ [.p,q.[ or t in [.(min (r,p)),(max (s,q)).[ ) & ( not t in [.(min (r,p)),(max (s,q)).[ or t in [.r,s.[ \/ [.p,q.[ ) )
thus ( t in [.r,s.[ \/ [.p,q.[ implies t in [.(min (r,p)),(max (s,q)).[ ) ::_thesis: ( not t in [.(min (r,p)),(max (s,q)).[ or t in [.r,s.[ \/ [.p,q.[ )
proof
assume t in [.r,s.[ \/ [.p,q.[ ; ::_thesis: t in [.(min (r,p)),(max (s,q)).[
then ( t in [.r,s.[ or t in [.p,q.[ ) by XBOOLE_0:def_3;
then A3: ( ( r <= t & t < s ) or ( p <= t & t < q ) ) by Th3;
then A4: min (r,p) <= t by XXREAL_0:23;
t < max (s,q) by A3, XXREAL_0:30;
hence t in [.(min (r,p)),(max (s,q)).[ by A4, Th3; ::_thesis: verum
end;
A5: r <= u by A1, Th3;
A6: u < s by A1, Th3;
A7: p <= u by A2, Th3;
A8: u < q by A2, Th3;
assume A9: t in [.(min (r,p)),(max (s,q)).[ ; ::_thesis: t in [.r,s.[ \/ [.p,q.[
then A10: min (r,p) <= t by Th3;
A11: t < max (s,q) by A9, Th3;
percases ( ( r <= t & t < s ) or ( p <= t & t < q ) or ( p <= t & t < s ) or ( r <= t & t < q ) ) by A10, A11, XXREAL_0:21, XXREAL_0:28;
suppose ( ( r <= t & t < s ) or ( p <= t & t < q ) ) ; ::_thesis: t in [.r,s.[ \/ [.p,q.[
then ( t in [.r,s.[ or t in [.p,q.[ ) by Th3;
hence t in [.r,s.[ \/ [.p,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
supposethat A12: p <= t and
A13: t < s ; ::_thesis: t in [.r,s.[ \/ [.p,q.[
( u <= t or t <= u ) ;
then ( r <= t or t < q ) by A5, A8, XXREAL_0:2;
then ( t in [.r,s.[ or t in [.p,q.[ ) by A12, A13, Th3;
hence t in [.r,s.[ \/ [.p,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
supposethat A14: r <= t and
A15: t < q ; ::_thesis: t in [.r,s.[ \/ [.p,q.[
( u <= t or t <= u ) ;
then ( t < s or p <= t ) by A6, A7, XXREAL_0:2;
then ( t in [.r,s.[ or t in [.p,q.[ ) by A14, A15, Th3;
hence t in [.r,s.[ \/ [.p,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:163
for r, s, p, q being ext-real number holds ].r,s.] \/ ].p,q.] c= ].(min (r,p)),(max (s,q)).]
proof
let r, s, p, q be ext-real number ; ::_thesis: ].r,s.] \/ ].p,q.] c= ].(min (r,p)),(max (s,q)).]
let t be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not t in ].r,s.] \/ ].p,q.] or t in ].(min (r,p)),(max (s,q)).] )
assume t in ].r,s.] \/ ].p,q.] ; ::_thesis: t in ].(min (r,p)),(max (s,q)).]
then ( t in ].r,s.] or t in ].p,q.] ) by XBOOLE_0:def_3;
then A1: ( ( r < t & t <= s ) or ( p < t & t <= q ) ) by Th2;
then A2: min (r,p) < t by XXREAL_0:22;
t <= max (s,q) by A1, XXREAL_0:31;
hence t in ].(min (r,p)),(max (s,q)).] by A2, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:164
for r, s, p, q being ext-real number st ].r,s.] meets ].p,q.] holds
].r,s.] \/ ].p,q.] = ].(min (r,p)),(max (s,q)).]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( ].r,s.] meets ].p,q.] implies ].r,s.] \/ ].p,q.] = ].(min (r,p)),(max (s,q)).] )
assume ].r,s.] meets ].p,q.] ; ::_thesis: ].r,s.] \/ ].p,q.] = ].(min (r,p)),(max (s,q)).]
then consider u being ext-real number such that
A1: u in ].r,s.] and
A2: u in ].p,q.] by MEMBERED:def_20;
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].r,s.] \/ ].p,q.] or t in ].(min (r,p)),(max (s,q)).] ) & ( not t in ].(min (r,p)),(max (s,q)).] or t in ].r,s.] \/ ].p,q.] ) )
thus ( t in ].r,s.] \/ ].p,q.] implies t in ].(min (r,p)),(max (s,q)).] ) ::_thesis: ( not t in ].(min (r,p)),(max (s,q)).] or t in ].r,s.] \/ ].p,q.] )
proof
assume t in ].r,s.] \/ ].p,q.] ; ::_thesis: t in ].(min (r,p)),(max (s,q)).]
then ( t in ].r,s.] or t in ].p,q.] ) by XBOOLE_0:def_3;
then A3: ( ( r < t & t <= s ) or ( p < t & t <= q ) ) by Th2;
then A4: min (r,p) < t by XXREAL_0:22;
t <= max (s,q) by A3, XXREAL_0:31;
hence t in ].(min (r,p)),(max (s,q)).] by A4, Th2; ::_thesis: verum
end;
A5: r < u by A1, Th2;
A6: u <= s by A1, Th2;
A7: p < u by A2, Th2;
A8: u <= q by A2, Th2;
assume A9: t in ].(min (r,p)),(max (s,q)).] ; ::_thesis: t in ].r,s.] \/ ].p,q.]
then A10: min (r,p) < t by Th2;
A11: t <= max (s,q) by A9, Th2;
percases ( ( r < t & t <= s ) or ( p < t & t <= q ) or ( p < t & t <= s ) or ( r < t & t <= q ) ) by A10, A11, XXREAL_0:20, XXREAL_0:29;
suppose ( ( r < t & t <= s ) or ( p < t & t <= q ) ) ; ::_thesis: t in ].r,s.] \/ ].p,q.]
then ( t in ].r,s.] or t in ].p,q.] ) by Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
supposethat A12: p < t and
A13: t <= s ; ::_thesis: t in ].r,s.] \/ ].p,q.]
( u <= t or t <= u ) ;
then ( r < t or t <= q ) by A5, A8, XXREAL_0:2;
then ( t in ].r,s.] or t in ].p,q.] ) by A12, A13, Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
supposethat A14: r < t and
A15: t <= q ; ::_thesis: t in ].r,s.] \/ ].p,q.]
( u <= t or t <= u ) ;
then ( t <= s or p < t ) by A6, A7, XXREAL_0:2;
then ( t in ].r,s.] or t in ].p,q.] ) by A14, A15, Th2;
hence t in ].r,s.] \/ ].p,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:165
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,s.] \/ [.s,t.] = [.r,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,s.] \/ [.s,t.] = [.r,t.] )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,s.] \/ [.s,t.] = [.r,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.] \/ [.s,t.] or p in [.r,t.] ) & ( not p in [.r,t.] or p in [.r,s.] \/ [.s,t.] ) )
thus ( p in [.r,s.] \/ [.s,t.] implies p in [.r,t.] ) ::_thesis: ( not p in [.r,t.] or p in [.r,s.] \/ [.s,t.] )
proof
assume p in [.r,s.] \/ [.s,t.] ; ::_thesis: p in [.r,t.]
then ( p in [.r,s.] or p in [.s,t.] ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p <= s ) or ( s <= p & p <= t ) ) by Th1;
then A4: r <= p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in [.r,t.] by A4, Th1; ::_thesis: verum
end;
assume p in [.r,t.] ; ::_thesis: p in [.r,s.] \/ [.s,t.]
then ( ( r <= p & p <= s ) or ( s <= p & p <= t ) ) by Th1;
then ( p in [.r,s.] or p in [.s,t.] ) by Th1;
hence p in [.r,s.] \/ [.s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th166: :: XXREAL_1:166
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,s.[ \/ [.s,t.] = [.r,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,s.[ \/ [.s,t.] = [.r,t.] )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,s.[ \/ [.s,t.] = [.r,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.[ \/ [.s,t.] or p in [.r,t.] ) & ( not p in [.r,t.] or p in [.r,s.[ \/ [.s,t.] ) )
thus ( p in [.r,s.[ \/ [.s,t.] implies p in [.r,t.] ) ::_thesis: ( not p in [.r,t.] or p in [.r,s.[ \/ [.s,t.] )
proof
assume p in [.r,s.[ \/ [.s,t.] ; ::_thesis: p in [.r,t.]
then ( p in [.r,s.[ or p in [.s,t.] ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p < s ) or ( s <= p & p <= t ) ) by Th1, Th3;
then A4: r <= p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in [.r,t.] by A4, Th1; ::_thesis: verum
end;
assume p in [.r,t.] ; ::_thesis: p in [.r,s.[ \/ [.s,t.]
then ( ( r <= p & p < s ) or ( s <= p & p <= t ) ) by Th1;
then ( p in [.r,s.[ or p in [.s,t.] ) by Th1, Th3;
hence p in [.r,s.[ \/ [.s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th167: :: XXREAL_1:167
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,s.] \/ ].s,t.] = [.r,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,s.] \/ ].s,t.] = [.r,t.] )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,s.] \/ ].s,t.] = [.r,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.] \/ ].s,t.] or p in [.r,t.] ) & ( not p in [.r,t.] or p in [.r,s.] \/ ].s,t.] ) )
thus ( p in [.r,s.] \/ ].s,t.] implies p in [.r,t.] ) ::_thesis: ( not p in [.r,t.] or p in [.r,s.] \/ ].s,t.] )
proof
assume p in [.r,s.] \/ ].s,t.] ; ::_thesis: p in [.r,t.]
then ( p in [.r,s.] or p in ].s,t.] ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p <= s ) or ( s < p & p <= t ) ) by Th1, Th2;
then A4: r <= p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in [.r,t.] by A4, Th1; ::_thesis: verum
end;
assume p in [.r,t.] ; ::_thesis: p in [.r,s.] \/ ].s,t.]
then ( ( r <= p & p <= s ) or ( s < p & p <= t ) ) by Th1;
then ( p in [.r,s.] or p in ].s,t.] ) by Th1, Th2;
hence p in [.r,s.] \/ ].s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:168
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,s.[ \/ [.s,t.[ = [.r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,s.[ \/ [.s,t.[ = [.r,t.[ )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,s.[ \/ [.s,t.[ = [.r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.[ \/ [.s,t.[ or p in [.r,t.[ ) & ( not p in [.r,t.[ or p in [.r,s.[ \/ [.s,t.[ ) )
thus ( p in [.r,s.[ \/ [.s,t.[ implies p in [.r,t.[ ) ::_thesis: ( not p in [.r,t.[ or p in [.r,s.[ \/ [.s,t.[ )
proof
assume p in [.r,s.[ \/ [.s,t.[ ; ::_thesis: p in [.r,t.[
then ( p in [.r,s.[ or p in [.s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p < s ) or ( s <= p & p < t ) ) by Th3;
then A4: r <= p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in [.r,t.[ by A4, Th3; ::_thesis: verum
end;
assume p in [.r,t.[ ; ::_thesis: p in [.r,s.[ \/ [.s,t.[
then ( ( r <= p & p < s ) or ( s <= p & p < t ) ) by Th3;
then ( p in [.r,s.[ or p in [.s,t.[ ) by Th3;
hence p in [.r,s.[ \/ [.s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th169: :: XXREAL_1:169
for r, s, t being ext-real number st r <= s & s < t holds
[.r,s.] \/ ].s,t.[ = [.r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s < t implies [.r,s.] \/ ].s,t.[ = [.r,t.[ )
assume that
A1: r <= s and
A2: s < t ; ::_thesis: [.r,s.] \/ ].s,t.[ = [.r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.] \/ ].s,t.[ or p in [.r,t.[ ) & ( not p in [.r,t.[ or p in [.r,s.] \/ ].s,t.[ ) )
thus ( p in [.r,s.] \/ ].s,t.[ implies p in [.r,t.[ ) ::_thesis: ( not p in [.r,t.[ or p in [.r,s.] \/ ].s,t.[ )
proof
assume p in [.r,s.] \/ ].s,t.[ ; ::_thesis: p in [.r,t.[
then ( p in [.r,s.] or p in ].s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p <= s ) or ( s < p & p < t ) ) by Th1, Th4;
then A4: r <= p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in [.r,t.[ by A4, Th3; ::_thesis: verum
end;
assume p in [.r,t.[ ; ::_thesis: p in [.r,s.] \/ ].s,t.[
then ( ( r <= p & p <= s ) or ( s < p & p < t ) ) by Th3;
then ( p in [.r,s.] or p in ].s,t.[ ) by Th1, Th4;
hence p in [.r,s.] \/ ].s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:170
for r, s, t being ext-real number st r <= s & s <= t holds
].r,s.] \/ ].s,t.] = ].r,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies ].r,s.] \/ ].s,t.] = ].r,t.] )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: ].r,s.] \/ ].s,t.] = ].r,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.] \/ ].s,t.] or p in ].r,t.] ) & ( not p in ].r,t.] or p in ].r,s.] \/ ].s,t.] ) )
thus ( p in ].r,s.] \/ ].s,t.] implies p in ].r,t.] ) ::_thesis: ( not p in ].r,t.] or p in ].r,s.] \/ ].s,t.] )
proof
assume p in ].r,s.] \/ ].s,t.] ; ::_thesis: p in ].r,t.]
then ( p in ].r,s.] or p in ].s,t.] ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p <= s ) or ( s < p & p <= t ) ) by Th2;
then A4: r < p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in ].r,t.] by A4, Th2; ::_thesis: verum
end;
assume p in ].r,t.] ; ::_thesis: p in ].r,s.] \/ ].s,t.]
then ( ( r < p & p <= s ) or ( s < p & p <= t ) ) by Th2;
then ( p in ].r,s.] or p in ].s,t.] ) by Th2;
hence p in ].r,s.] \/ ].s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th171: :: XXREAL_1:171
for r, s, t being ext-real number st r <= s & s < t holds
].r,s.] \/ ].s,t.[ = ].r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s < t implies ].r,s.] \/ ].s,t.[ = ].r,t.[ )
assume that
A1: r <= s and
A2: s < t ; ::_thesis: ].r,s.] \/ ].s,t.[ = ].r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.] \/ ].s,t.[ or p in ].r,t.[ ) & ( not p in ].r,t.[ or p in ].r,s.] \/ ].s,t.[ ) )
thus ( p in ].r,s.] \/ ].s,t.[ implies p in ].r,t.[ ) ::_thesis: ( not p in ].r,t.[ or p in ].r,s.] \/ ].s,t.[ )
proof
assume p in ].r,s.] \/ ].s,t.[ ; ::_thesis: p in ].r,t.[
then ( p in ].r,s.] or p in ].s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p <= s ) or ( s < p & p < t ) ) by Th2, Th4;
then A4: r < p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in ].r,t.[ by A4, Th4; ::_thesis: verum
end;
assume p in ].r,t.[ ; ::_thesis: p in ].r,s.] \/ ].s,t.[
then ( ( r < p & p <= s ) or ( s < p & p < t ) ) by Th4;
then ( p in ].r,s.] or p in ].s,t.[ ) by Th2, Th4;
hence p in ].r,s.] \/ ].s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:172
for r, s, t being ext-real number st r < s & s < t holds
].r,s.] \/ [.s,t.[ = ].r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s < t implies ].r,s.] \/ [.s,t.[ = ].r,t.[ )
assume that
A1: r < s and
A2: s < t ; ::_thesis: ].r,s.] \/ [.s,t.[ = ].r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.] \/ [.s,t.[ or p in ].r,t.[ ) & ( not p in ].r,t.[ or p in ].r,s.] \/ [.s,t.[ ) )
thus ( p in ].r,s.] \/ [.s,t.[ implies p in ].r,t.[ ) ::_thesis: ( not p in ].r,t.[ or p in ].r,s.] \/ [.s,t.[ )
proof
assume p in ].r,s.] \/ [.s,t.[ ; ::_thesis: p in ].r,t.[
then ( p in ].r,s.] or p in [.s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p <= s ) or ( s <= p & p < t ) ) by Th2, Th3;
then A4: r < p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in ].r,t.[ by A4, Th4; ::_thesis: verum
end;
assume p in ].r,t.[ ; ::_thesis: p in ].r,s.] \/ [.s,t.[
then ( ( r < p & p <= s ) or ( s < p & p < t ) ) by Th4;
then ( p in ].r,s.] or p in [.s,t.[ ) by Th2, Th3;
hence p in ].r,s.] \/ [.s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th173: :: XXREAL_1:173
for r, s, t being ext-real number st r < s & s < t holds
].r,s.[ \/ [.s,t.[ = ].r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s < t implies ].r,s.[ \/ [.s,t.[ = ].r,t.[ )
assume that
A1: r < s and
A2: s < t ; ::_thesis: ].r,s.[ \/ [.s,t.[ = ].r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.[ \/ [.s,t.[ or p in ].r,t.[ ) & ( not p in ].r,t.[ or p in ].r,s.[ \/ [.s,t.[ ) )
thus ( p in ].r,s.[ \/ [.s,t.[ implies p in ].r,t.[ ) ::_thesis: ( not p in ].r,t.[ or p in ].r,s.[ \/ [.s,t.[ )
proof
assume p in ].r,s.[ \/ [.s,t.[ ; ::_thesis: p in ].r,t.[
then ( p in ].r,s.[ or p in [.s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p < s ) or ( s <= p & p < t ) ) by Th3, Th4;
then A4: r < p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in ].r,t.[ by A4, Th4; ::_thesis: verum
end;
assume p in ].r,t.[ ; ::_thesis: p in ].r,s.[ \/ [.s,t.[
then ( ( r < p & p < s ) or ( s <= p & p < t ) ) by Th4;
then ( p in ].r,s.[ or p in [.s,t.[ ) by Th3, Th4;
hence p in ].r,s.[ \/ [.s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th174: :: XXREAL_1:174
for p, s, r, q being ext-real number st p <= s & r <= q & s <= r holds
[.p,r.] \/ [.s,q.] = [.p,q.]
proof
let p, s, r, q be ext-real number ; ::_thesis: ( p <= s & r <= q & s <= r implies [.p,r.] \/ [.s,q.] = [.p,q.] )
assume that
A1: p <= s and
A2: r <= q and
A3: s <= r ; ::_thesis: [.p,r.] \/ [.s,q.] = [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.p,r.] \/ [.s,q.] or t in [.p,q.] ) & ( not t in [.p,q.] or t in [.p,r.] \/ [.s,q.] ) )
thus ( t in [.p,r.] \/ [.s,q.] implies t in [.p,q.] ) ::_thesis: ( not t in [.p,q.] or t in [.p,r.] \/ [.s,q.] )
proof
assume t in [.p,r.] \/ [.s,q.] ; ::_thesis: t in [.p,q.]
then ( t in [.p,r.] or t in [.s,q.] ) by XBOOLE_0:def_3;
then A4: ( ( p <= t & t <= r ) or ( s <= t & t <= q ) ) by Th1;
then A5: p <= t by A1, XXREAL_0:2;
t <= q by A2, A4, XXREAL_0:2;
hence t in [.p,q.] by A5, Th1; ::_thesis: verum
end;
assume t in [.p,q.] ; ::_thesis: t in [.p,r.] \/ [.s,q.]
then ( ( p <= t & t <= r ) or ( s <= t & t <= q ) ) by A3, Th1, XXREAL_0:2;
then ( t in [.p,r.] or t in [.s,q.] ) by Th1;
hence t in [.p,r.] \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th175: :: XXREAL_1:175
for p, s, r, q being ext-real number st p <= s & r <= q & s < r holds
[.p,r.[ \/ ].s,q.] = [.p,q.]
proof
let p, s, r, q be ext-real number ; ::_thesis: ( p <= s & r <= q & s < r implies [.p,r.[ \/ ].s,q.] = [.p,q.] )
assume that
A1: p <= s and
A2: r <= q and
A3: s < r ; ::_thesis: [.p,r.[ \/ ].s,q.] = [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.p,r.[ \/ ].s,q.] or t in [.p,q.] ) & ( not t in [.p,q.] or t in [.p,r.[ \/ ].s,q.] ) )
thus ( t in [.p,r.[ \/ ].s,q.] implies t in [.p,q.] ) ::_thesis: ( not t in [.p,q.] or t in [.p,r.[ \/ ].s,q.] )
proof
assume t in [.p,r.[ \/ ].s,q.] ; ::_thesis: t in [.p,q.]
then ( t in [.p,r.[ or t in ].s,q.] ) by XBOOLE_0:def_3;
then A4: ( ( p <= t & t <= r ) or ( s <= t & t <= q ) ) by Th2, Th3;
then A5: p <= t by A1, XXREAL_0:2;
t <= q by A2, A4, XXREAL_0:2;
hence t in [.p,q.] by A5, Th1; ::_thesis: verum
end;
assume t in [.p,q.] ; ::_thesis: t in [.p,r.[ \/ ].s,q.]
then ( ( p <= t & t < r ) or ( s < t & t <= q ) ) by A3, Th1, XXREAL_0:2;
then ( t in [.p,r.[ or t in ].s,q.] ) by Th2, Th3;
hence t in [.p,r.[ \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:176
for p, s, r, q being ext-real number st p <= s & s <= r & r < q holds
[.p,r.] \/ [.s,q.[ = [.p,q.[
proof
let p, s, r, q be ext-real number ; ::_thesis: ( p <= s & s <= r & r < q implies [.p,r.] \/ [.s,q.[ = [.p,q.[ )
assume that
A1: p <= s and
A2: s <= r and
A3: r < q ; ::_thesis: [.p,r.] \/ [.s,q.[ = [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.p,r.] \/ [.s,q.[ or t in [.p,q.[ ) & ( not t in [.p,q.[ or t in [.p,r.] \/ [.s,q.[ ) )
thus ( t in [.p,r.] \/ [.s,q.[ implies t in [.p,q.[ ) ::_thesis: ( not t in [.p,q.[ or t in [.p,r.] \/ [.s,q.[ )
proof
assume t in [.p,r.] \/ [.s,q.[ ; ::_thesis: t in [.p,q.[
then ( t in [.p,r.] or t in [.s,q.[ ) by XBOOLE_0:def_3;
then A4: ( ( p <= t & t <= r ) or ( s <= t & t < q ) ) by Th1, Th3;
then A5: p <= t by A1, XXREAL_0:2;
t < q by A3, A4, XXREAL_0:2;
hence t in [.p,q.[ by A5, Th3; ::_thesis: verum
end;
assume t in [.p,q.[ ; ::_thesis: t in [.p,r.] \/ [.s,q.[
then ( ( p <= t & t <= r ) or ( s <= t & t < q ) ) by A2, Th3, XXREAL_0:2;
then ( t in [.p,r.] or t in [.s,q.[ ) by Th1, Th3;
hence t in [.p,r.] \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:177
for p, s, r, q being ext-real number st p < s & r <= q & s <= r holds
].p,r.] \/ [.s,q.] = ].p,q.]
proof
let p, s, r, q be ext-real number ; ::_thesis: ( p < s & r <= q & s <= r implies ].p,r.] \/ [.s,q.] = ].p,q.] )
assume that
A1: p < s and
A2: r <= q and
A3: s <= r ; ::_thesis: ].p,r.] \/ [.s,q.] = ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].p,r.] \/ [.s,q.] or t in ].p,q.] ) & ( not t in ].p,q.] or t in ].p,r.] \/ [.s,q.] ) )
thus ( t in ].p,r.] \/ [.s,q.] implies t in ].p,q.] ) ::_thesis: ( not t in ].p,q.] or t in ].p,r.] \/ [.s,q.] )
proof
assume t in ].p,r.] \/ [.s,q.] ; ::_thesis: t in ].p,q.]
then ( t in ].p,r.] or t in [.s,q.] ) by XBOOLE_0:def_3;
then A4: ( ( p < t & t <= r ) or ( s <= t & t <= q ) ) by Th1, Th2;
then A5: p < t by A1, XXREAL_0:2;
t <= q by A2, A4, XXREAL_0:2;
hence t in ].p,q.] by A5, Th2; ::_thesis: verum
end;
assume t in ].p,q.] ; ::_thesis: t in ].p,r.] \/ [.s,q.]
then ( ( p < t & t <= r ) or ( s <= t & t <= q ) ) by A3, Th2, XXREAL_0:2;
then ( t in ].p,r.] or t in [.s,q.] ) by Th1, Th2;
hence t in ].p,r.] \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem Th178: :: XXREAL_1:178
for p, s, r, q being ext-real number st p < s & r < q & s <= r holds
].p,r.] \/ [.s,q.[ = ].p,q.[
proof
let p, s, r, q be ext-real number ; ::_thesis: ( p < s & r < q & s <= r implies ].p,r.] \/ [.s,q.[ = ].p,q.[ )
assume that
A1: p < s and
A2: r < q and
A3: s <= r ; ::_thesis: ].p,r.] \/ [.s,q.[ = ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].p,r.] \/ [.s,q.[ or t in ].p,q.[ ) & ( not t in ].p,q.[ or t in ].p,r.] \/ [.s,q.[ ) )
thus ( t in ].p,r.] \/ [.s,q.[ implies t in ].p,q.[ ) ::_thesis: ( not t in ].p,q.[ or t in ].p,r.] \/ [.s,q.[ )
proof
assume t in ].p,r.] \/ [.s,q.[ ; ::_thesis: t in ].p,q.[
then ( t in ].p,r.] or t in [.s,q.[ ) by XBOOLE_0:def_3;
then A4: ( ( p < t & t <= r ) or ( s <= t & t < q ) ) by Th2, Th3;
then A5: p < t by A1, XXREAL_0:2;
t < q by A2, A4, XXREAL_0:2;
hence t in ].p,q.[ by A5, Th4; ::_thesis: verum
end;
assume t in ].p,q.[ ; ::_thesis: t in ].p,r.] \/ [.s,q.[
then ( ( p < t & t <= r ) or ( s <= t & t < q ) ) by A3, Th4, XXREAL_0:2;
then ( t in ].p,r.] or t in [.s,q.[ ) by Th2, Th3;
hence t in ].p,r.] \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:179
for p, r, s, q being ext-real number st p <= r & p <= s & r <= q & s <= q holds
([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & p <= s & r <= q & s <= q implies ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,q.] )
assume that
A1: p <= r and
A2: p <= s and
A3: r <= q and
A4: s <= q ; ::_thesis: ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,q.]
percases ( r <= s or s < r ) ;
suppose r <= s ; ::_thesis: ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,q.]
hence ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,s.] \/ ].s,q.] by A1, Th166
.= [.p,q.] by A2, A4, Th167 ;
::_thesis: verum
end;
supposeA5: s < r ; ::_thesis: ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = [.p,q.]
hence ([.p,r.[ \/ [.r,s.]) \/ ].s,q.] = ([.p,r.[ \/ {}) \/ ].s,q.] by Th29
.= [.p,q.] by A2, A3, A5, Th175 ;
::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:180
for p, r, s, q being ext-real number st p < r & p < s & r < q & s < q holds
(].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,q.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p < r & p < s & r < q & s < q implies (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,q.[ )
assume that
A1: p < r and
A2: p < s and
A3: r < q and
A4: s < q ; ::_thesis: (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,q.[
percases ( r < s or s <= r ) ;
suppose r < s ; ::_thesis: (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,q.[
hence (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,s.[ \/ [.s,q.[ by A1, Th171
.= ].p,q.[ by A2, A4, Th173 ;
::_thesis: verum
end;
supposeA5: s <= r ; ::_thesis: (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = ].p,q.[
hence (].p,r.] \/ ].r,s.[) \/ [.s,q.[ = (].p,r.] \/ {}) \/ [.s,q.[ by Th28
.= ].p,q.[ by A2, A3, A5, Th178 ;
::_thesis: verum
end;
end;
end;
theorem :: XXREAL_1:181
for p, r, s, q being ext-real number st p <= r & r <= s & s <= q holds
([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,q.]
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & r <= s & s <= q implies ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,q.] )
assume that
A1: p <= r and
A2: r <= s and
A3: s <= q ; ::_thesis: ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,q.]
A4: p <= s by A1, A2, XXREAL_0:2;
A5: r <= q by A2, A3, XXREAL_0:2;
percases ( r < s or s <= r ) ;
suppose r < s ; ::_thesis: ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,q.]
hence ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,s.[ \/ [.s,q.] by A1, Th169
.= [.p,q.] by A3, A4, Th166 ;
::_thesis: verum
end;
supposeA6: s <= r ; ::_thesis: ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = [.p,q.]
hence ([.p,r.] \/ ].r,s.[) \/ [.s,q.] = ([.p,r.] \/ {}) \/ [.s,q.] by Th28
.= [.p,q.] by A4, A5, A6, Th174 ;
::_thesis: verum
end;
end;
end;
theorem Th182: :: XXREAL_1:182
for r, s, t being ext-real number st r <= s holds
[.r,t.] \ [.r,s.] = ].s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s implies [.r,t.] \ [.r,s.] = ].s,t.] )
assume A1: r <= s ; ::_thesis: [.r,t.] \ [.r,s.] = ].s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ [.r,s.] or p in ].s,t.] ) & ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] ) )
thus ( p in [.r,t.] \ [.r,s.] implies p in ].s,t.] ) ::_thesis: ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] )
proof
assume A2: p in [.r,t.] \ [.r,s.] ; ::_thesis: p in ].s,t.]
then A3: not p in [.r,s.] by XBOOLE_0:def_5;
A4: p <= t by A2, Th1;
( p < r or s < p ) by A3, Th1;
hence p in ].s,t.] by A2, A4, Th1, Th2; ::_thesis: verum
end;
assume A5: p in ].s,t.] ; ::_thesis: p in [.r,t.] \ [.r,s.]
then A6: s < p by Th2;
then A7: r <= p by A1, XXREAL_0:2;
p <= t by A5, Th2;
then A8: p in [.r,t.] by A7, Th1;
not p in [.r,s.] by A6, Th1;
hence p in [.r,t.] \ [.r,s.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th183: :: XXREAL_1:183
for r, s, t being ext-real number st r <= s holds
[.r,t.[ \ [.r,s.] = ].s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s implies [.r,t.[ \ [.r,s.] = ].s,t.[ )
assume A1: r <= s ; ::_thesis: [.r,t.[ \ [.r,s.] = ].s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.[ \ [.r,s.] or p in ].s,t.[ ) & ( not p in ].s,t.[ or p in [.r,t.[ \ [.r,s.] ) )
thus ( p in [.r,t.[ \ [.r,s.] implies p in ].s,t.[ ) ::_thesis: ( not p in ].s,t.[ or p in [.r,t.[ \ [.r,s.] )
proof
assume A2: p in [.r,t.[ \ [.r,s.] ; ::_thesis: p in ].s,t.[
then A3: not p in [.r,s.] by XBOOLE_0:def_5;
A4: p < t by A2, Th3;
( p < r or s < p ) by A3, Th1;
hence p in ].s,t.[ by A2, A4, Th3, Th4; ::_thesis: verum
end;
assume A5: p in ].s,t.[ ; ::_thesis: p in [.r,t.[ \ [.r,s.]
then A6: s < p by Th4;
then A7: r <= p by A1, XXREAL_0:2;
p < t by A5, Th4;
then A8: p in [.r,t.[ by A7, Th3;
not p in [.r,s.] by A6, Th1;
hence p in [.r,t.[ \ [.r,s.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th184: :: XXREAL_1:184
for r, s, t being ext-real number st r < s holds
[.r,t.] \ [.r,s.[ = [.s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s implies [.r,t.] \ [.r,s.[ = [.s,t.] )
assume A1: r < s ; ::_thesis: [.r,t.] \ [.r,s.[ = [.s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ [.r,s.[ or p in [.s,t.] ) & ( not p in [.s,t.] or p in [.r,t.] \ [.r,s.[ ) )
thus ( p in [.r,t.] \ [.r,s.[ implies p in [.s,t.] ) ::_thesis: ( not p in [.s,t.] or p in [.r,t.] \ [.r,s.[ )
proof
assume A2: p in [.r,t.] \ [.r,s.[ ; ::_thesis: p in [.s,t.]
then A3: not p in [.r,s.[ by XBOOLE_0:def_5;
A4: p <= t by A2, Th1;
( p < r or s <= p ) by A3, Th3;
hence p in [.s,t.] by A2, A4, Th1; ::_thesis: verum
end;
assume A5: p in [.s,t.] ; ::_thesis: p in [.r,t.] \ [.r,s.[
then A6: s <= p by Th1;
then A7: r <= p by A1, XXREAL_0:2;
p <= t by A5, Th1;
then A8: p in [.r,t.] by A7, Th1;
not p in [.r,s.[ by A6, Th3;
hence p in [.r,t.] \ [.r,s.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th185: :: XXREAL_1:185
for r, s, t being ext-real number st r < s holds
[.r,t.[ \ [.r,s.[ = [.s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s implies [.r,t.[ \ [.r,s.[ = [.s,t.[ )
assume A1: r < s ; ::_thesis: [.r,t.[ \ [.r,s.[ = [.s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.[ \ [.r,s.[ or p in [.s,t.[ ) & ( not p in [.s,t.[ or p in [.r,t.[ \ [.r,s.[ ) )
thus ( p in [.r,t.[ \ [.r,s.[ implies p in [.s,t.[ ) ::_thesis: ( not p in [.s,t.[ or p in [.r,t.[ \ [.r,s.[ )
proof
assume A2: p in [.r,t.[ \ [.r,s.[ ; ::_thesis: p in [.s,t.[
then A3: not p in [.r,s.[ by XBOOLE_0:def_5;
A4: p < t by A2, Th3;
( p < r or s <= p ) by A3, Th3;
hence p in [.s,t.[ by A2, A4, Th3; ::_thesis: verum
end;
assume A5: p in [.s,t.[ ; ::_thesis: p in [.r,t.[ \ [.r,s.[
then A6: s <= p by Th3;
then A7: r <= p by A1, XXREAL_0:2;
p < t by A5, Th3;
then A8: p in [.r,t.[ by A7, Th3;
not p in [.r,s.[ by A6, Th3;
hence p in [.r,t.[ \ [.r,s.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th186: :: XXREAL_1:186
for r, s, t being ext-real number st r <= s holds
[.r,t.] \ [.r,s.] = ].s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s implies [.r,t.] \ [.r,s.] = ].s,t.] )
assume A1: r <= s ; ::_thesis: [.r,t.] \ [.r,s.] = ].s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ [.r,s.] or p in ].s,t.] ) & ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] ) )
thus ( p in [.r,t.] \ [.r,s.] implies p in ].s,t.] ) ::_thesis: ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] )
proof
assume A2: p in [.r,t.] \ [.r,s.] ; ::_thesis: p in ].s,t.]
then A3: not p in [.r,s.] by XBOOLE_0:def_5;
A4: p <= t by A2, Th1;
( p < r or s < p ) by A3, Th1;
hence p in ].s,t.] by A2, A4, Th1, Th2; ::_thesis: verum
end;
assume A5: p in ].s,t.] ; ::_thesis: p in [.r,t.] \ [.r,s.]
then A6: s < p by Th2;
then A7: r <= p by A1, XXREAL_0:2;
p <= t by A5, Th2;
then A8: p in [.r,t.] by A7, Th1;
not p in [.r,s.] by A6, Th1;
hence p in [.r,t.] \ [.r,s.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th187: :: XXREAL_1:187
for r, s, t being ext-real number st r < s holds
].r,t.[ \ ].r,s.] = ].s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s implies ].r,t.[ \ ].r,s.] = ].s,t.[ )
assume A1: r < s ; ::_thesis: ].r,t.[ \ ].r,s.] = ].s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.[ \ ].r,s.] or p in ].s,t.[ ) & ( not p in ].s,t.[ or p in ].r,t.[ \ ].r,s.] ) )
thus ( p in ].r,t.[ \ ].r,s.] implies p in ].s,t.[ ) ::_thesis: ( not p in ].s,t.[ or p in ].r,t.[ \ ].r,s.] )
proof
assume A2: p in ].r,t.[ \ ].r,s.] ; ::_thesis: p in ].s,t.[
then A3: not p in ].r,s.] by XBOOLE_0:def_5;
A4: p < t by A2, Th4;
( p <= r or s < p ) by A3, Th2;
hence p in ].s,t.[ by A2, A4, Th4; ::_thesis: verum
end;
assume A5: p in ].s,t.[ ; ::_thesis: p in ].r,t.[ \ ].r,s.]
then A6: s < p by Th4;
then A7: r < p by A1, XXREAL_0:2;
p < t by A5, Th4;
then A8: p in ].r,t.[ by A7, Th4;
not p in ].r,s.] by A6, Th2;
hence p in ].r,t.[ \ ].r,s.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th188: :: XXREAL_1:188
for r, s, t being ext-real number st r < s holds
].r,t.] \ ].r,s.[ = [.s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s implies ].r,t.] \ ].r,s.[ = [.s,t.] )
assume A1: r < s ; ::_thesis: ].r,t.] \ ].r,s.[ = [.s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.] \ ].r,s.[ or p in [.s,t.] ) & ( not p in [.s,t.] or p in ].r,t.] \ ].r,s.[ ) )
thus ( p in ].r,t.] \ ].r,s.[ implies p in [.s,t.] ) ::_thesis: ( not p in [.s,t.] or p in ].r,t.] \ ].r,s.[ )
proof
assume A2: p in ].r,t.] \ ].r,s.[ ; ::_thesis: p in [.s,t.]
then A3: not p in ].r,s.[ by XBOOLE_0:def_5;
A4: p <= t by A2, Th2;
( p <= r or s <= p ) by A3, Th4;
hence p in [.s,t.] by A2, A4, Th1, Th2; ::_thesis: verum
end;
assume A5: p in [.s,t.] ; ::_thesis: p in ].r,t.] \ ].r,s.[
then A6: s <= p by Th1;
then A7: r < p by A1, XXREAL_0:2;
p <= t by A5, Th1;
then A8: p in ].r,t.] by A7, Th2;
not p in ].r,s.[ by A6, Th4;
hence p in ].r,t.] \ ].r,s.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th189: :: XXREAL_1:189
for r, s, t being ext-real number st r < s holds
].r,t.[ \ ].r,s.[ = [.s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s implies ].r,t.[ \ ].r,s.[ = [.s,t.[ )
assume A1: r < s ; ::_thesis: ].r,t.[ \ ].r,s.[ = [.s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.[ \ ].r,s.[ or p in [.s,t.[ ) & ( not p in [.s,t.[ or p in ].r,t.[ \ ].r,s.[ ) )
thus ( p in ].r,t.[ \ ].r,s.[ implies p in [.s,t.[ ) ::_thesis: ( not p in [.s,t.[ or p in ].r,t.[ \ ].r,s.[ )
proof
assume A2: p in ].r,t.[ \ ].r,s.[ ; ::_thesis: p in [.s,t.[
then A3: not p in ].r,s.[ by XBOOLE_0:def_5;
A4: p < t by A2, Th4;
( p <= r or s <= p ) by A3, Th4;
hence p in [.s,t.[ by A2, A4, Th3, Th4; ::_thesis: verum
end;
assume A5: p in [.s,t.[ ; ::_thesis: p in ].r,t.[ \ ].r,s.[
then A6: s <= p by Th3;
then A7: r < p by A1, XXREAL_0:2;
p < t by A5, Th3;
then A8: p in ].r,t.[ by A7, Th4;
not p in ].r,s.[ by A6, Th4;
hence p in ].r,t.[ \ ].r,s.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th190: :: XXREAL_1:190
for s, t, r being ext-real number st s <= t holds
[.r,t.] \ [.s,t.] = [.r,s.[
proof
let s, t, r be ext-real number ; ::_thesis: ( s <= t implies [.r,t.] \ [.s,t.] = [.r,s.[ )
assume A1: s <= t ; ::_thesis: [.r,t.] \ [.s,t.] = [.r,s.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ [.s,t.] or p in [.r,s.[ ) & ( not p in [.r,s.[ or p in [.r,t.] \ [.s,t.] ) )
thus ( p in [.r,t.] \ [.s,t.] implies p in [.r,s.[ ) ::_thesis: ( not p in [.r,s.[ or p in [.r,t.] \ [.s,t.] )
proof
assume A2: p in [.r,t.] \ [.s,t.] ; ::_thesis: p in [.r,s.[
then A3: not p in [.s,t.] by XBOOLE_0:def_5;
A4: r <= p by A2, Th1;
( p < s or t < p ) by A3, Th1;
hence p in [.r,s.[ by A2, A4, Th1, Th3; ::_thesis: verum
end;
assume A5: p in [.r,s.[ ; ::_thesis: p in [.r,t.] \ [.s,t.]
then A6: p < s by Th3;
A7: r <= p by A5, Th3;
p <= t by A1, A6, XXREAL_0:2;
then A8: p in [.r,t.] by A7, Th1;
not p in [.s,t.] by A6, Th1;
hence p in [.r,t.] \ [.s,t.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th191: :: XXREAL_1:191
for s, t, r being ext-real number st s <= t holds
].r,t.] \ [.s,t.] = ].r,s.[
proof
let s, t, r be ext-real number ; ::_thesis: ( s <= t implies ].r,t.] \ [.s,t.] = ].r,s.[ )
assume A1: s <= t ; ::_thesis: ].r,t.] \ [.s,t.] = ].r,s.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.] \ [.s,t.] or p in ].r,s.[ ) & ( not p in ].r,s.[ or p in ].r,t.] \ [.s,t.] ) )
thus ( p in ].r,t.] \ [.s,t.] implies p in ].r,s.[ ) ::_thesis: ( not p in ].r,s.[ or p in ].r,t.] \ [.s,t.] )
proof
assume A2: p in ].r,t.] \ [.s,t.] ; ::_thesis: p in ].r,s.[
then A3: not p in [.s,t.] by XBOOLE_0:def_5;
A4: r < p by A2, Th2;
( p < s or t < p ) by A3, Th1;
hence p in ].r,s.[ by A2, A4, Th2, Th4; ::_thesis: verum
end;
assume A5: p in ].r,s.[ ; ::_thesis: p in ].r,t.] \ [.s,t.]
then A6: p < s by Th4;
A7: r < p by A5, Th4;
p <= t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.] by A7, Th2;
not p in [.s,t.] by A6, Th1;
hence p in ].r,t.] \ [.s,t.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th192: :: XXREAL_1:192
for s, t, r being ext-real number st s < t holds
[.r,t.] \ ].s,t.] = [.r,s.]
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies [.r,t.] \ ].s,t.] = [.r,s.] )
assume A1: s < t ; ::_thesis: [.r,t.] \ ].s,t.] = [.r,s.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ ].s,t.] or p in [.r,s.] ) & ( not p in [.r,s.] or p in [.r,t.] \ ].s,t.] ) )
thus ( p in [.r,t.] \ ].s,t.] implies p in [.r,s.] ) ::_thesis: ( not p in [.r,s.] or p in [.r,t.] \ ].s,t.] )
proof
assume A2: p in [.r,t.] \ ].s,t.] ; ::_thesis: p in [.r,s.]
then A3: not p in ].s,t.] by XBOOLE_0:def_5;
A4: r <= p by A2, Th1;
( p <= s or t < p ) by A3, Th2;
hence p in [.r,s.] by A2, A4, Th1; ::_thesis: verum
end;
assume A5: p in [.r,s.] ; ::_thesis: p in [.r,t.] \ ].s,t.]
then A6: p <= s by Th1;
A7: r <= p by A5, Th1;
p <= t by A1, A6, XXREAL_0:2;
then A8: p in [.r,t.] by A7, Th1;
not p in ].s,t.] by A6, Th2;
hence p in [.r,t.] \ ].s,t.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th193: :: XXREAL_1:193
for s, t, r being ext-real number st s < t holds
].r,t.] \ ].s,t.] = ].r,s.]
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies ].r,t.] \ ].s,t.] = ].r,s.] )
assume A1: s < t ; ::_thesis: ].r,t.] \ ].s,t.] = ].r,s.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.] \ ].s,t.] or p in ].r,s.] ) & ( not p in ].r,s.] or p in ].r,t.] \ ].s,t.] ) )
thus ( p in ].r,t.] \ ].s,t.] implies p in ].r,s.] ) ::_thesis: ( not p in ].r,s.] or p in ].r,t.] \ ].s,t.] )
proof
assume A2: p in ].r,t.] \ ].s,t.] ; ::_thesis: p in ].r,s.]
then A3: not p in ].s,t.] by XBOOLE_0:def_5;
A4: r < p by A2, Th2;
( p <= s or t < p ) by A3, Th2;
hence p in ].r,s.] by A2, A4, Th2; ::_thesis: verum
end;
assume A5: p in ].r,s.] ; ::_thesis: p in ].r,t.] \ ].s,t.]
then A6: p <= s by Th2;
A7: r < p by A5, Th2;
p <= t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.] by A7, Th2;
not p in ].s,t.] by A6, Th2;
hence p in ].r,t.] \ ].s,t.] by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th194: :: XXREAL_1:194
for s, t, r being ext-real number st s < t holds
[.r,t.[ \ [.s,t.[ = [.r,s.[
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies [.r,t.[ \ [.s,t.[ = [.r,s.[ )
assume A1: s < t ; ::_thesis: [.r,t.[ \ [.s,t.[ = [.r,s.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.[ \ [.s,t.[ or p in [.r,s.[ ) & ( not p in [.r,s.[ or p in [.r,t.[ \ [.s,t.[ ) )
thus ( p in [.r,t.[ \ [.s,t.[ implies p in [.r,s.[ ) ::_thesis: ( not p in [.r,s.[ or p in [.r,t.[ \ [.s,t.[ )
proof
assume A2: p in [.r,t.[ \ [.s,t.[ ; ::_thesis: p in [.r,s.[
then A3: not p in [.s,t.[ by XBOOLE_0:def_5;
A4: r <= p by A2, Th3;
( p < s or t <= p ) by A3, Th3;
hence p in [.r,s.[ by A2, A4, Th3; ::_thesis: verum
end;
assume A5: p in [.r,s.[ ; ::_thesis: p in [.r,t.[ \ [.s,t.[
then A6: p < s by Th3;
A7: r <= p by A5, Th3;
p < t by A1, A6, XXREAL_0:2;
then A8: p in [.r,t.[ by A7, Th3;
not p in [.s,t.[ by A6, Th3;
hence p in [.r,t.[ \ [.s,t.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th195: :: XXREAL_1:195
for s, t, r being ext-real number st s < t holds
].r,t.[ \ [.s,t.[ = ].r,s.[
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies ].r,t.[ \ [.s,t.[ = ].r,s.[ )
assume A1: s < t ; ::_thesis: ].r,t.[ \ [.s,t.[ = ].r,s.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.[ \ [.s,t.[ or p in ].r,s.[ ) & ( not p in ].r,s.[ or p in ].r,t.[ \ [.s,t.[ ) )
thus ( p in ].r,t.[ \ [.s,t.[ implies p in ].r,s.[ ) ::_thesis: ( not p in ].r,s.[ or p in ].r,t.[ \ [.s,t.[ )
proof
assume A2: p in ].r,t.[ \ [.s,t.[ ; ::_thesis: p in ].r,s.[
then A3: not p in [.s,t.[ by XBOOLE_0:def_5;
A4: r < p by A2, Th4;
( p < s or t <= p ) by A3, Th3;
hence p in ].r,s.[ by A2, A4, Th4; ::_thesis: verum
end;
assume A5: p in ].r,s.[ ; ::_thesis: p in ].r,t.[ \ [.s,t.[
then A6: p < s by Th4;
A7: r < p by A5, Th4;
p < t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.[ by A7, Th4;
not p in [.s,t.[ by A6, Th3;
hence p in ].r,t.[ \ [.s,t.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th196: :: XXREAL_1:196
for s, t, r being ext-real number st s < t holds
[.r,t.[ \ ].s,t.[ = [.r,s.]
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies [.r,t.[ \ ].s,t.[ = [.r,s.] )
assume A1: s < t ; ::_thesis: [.r,t.[ \ ].s,t.[ = [.r,s.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.[ \ ].s,t.[ or p in [.r,s.] ) & ( not p in [.r,s.] or p in [.r,t.[ \ ].s,t.[ ) )
thus ( p in [.r,t.[ \ ].s,t.[ implies p in [.r,s.] ) ::_thesis: ( not p in [.r,s.] or p in [.r,t.[ \ ].s,t.[ )
proof
assume A2: p in [.r,t.[ \ ].s,t.[ ; ::_thesis: p in [.r,s.]
then A3: not p in ].s,t.[ by XBOOLE_0:def_5;
A4: r <= p by A2, Th3;
( p <= s or t <= p ) by A3, Th4;
hence p in [.r,s.] by A2, A4, Th1, Th3; ::_thesis: verum
end;
assume A5: p in [.r,s.] ; ::_thesis: p in [.r,t.[ \ ].s,t.[
then A6: p <= s by Th1;
A7: r <= p by A5, Th1;
p < t by A1, A6, XXREAL_0:2;
then A8: p in [.r,t.[ by A7, Th3;
not p in ].s,t.[ by A6, Th4;
hence p in [.r,t.[ \ ].s,t.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th197: :: XXREAL_1:197
for s, t, r being ext-real number st s < t holds
].r,t.[ \ ].s,t.[ = ].r,s.]
proof
let s, t, r be ext-real number ; ::_thesis: ( s < t implies ].r,t.[ \ ].s,t.[ = ].r,s.] )
assume A1: s < t ; ::_thesis: ].r,t.[ \ ].s,t.[ = ].r,s.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.[ \ ].s,t.[ or p in ].r,s.] ) & ( not p in ].r,s.] or p in ].r,t.[ \ ].s,t.[ ) )
thus ( p in ].r,t.[ \ ].s,t.[ implies p in ].r,s.] ) ::_thesis: ( not p in ].r,s.] or p in ].r,t.[ \ ].s,t.[ )
proof
assume A2: p in ].r,t.[ \ ].s,t.[ ; ::_thesis: p in ].r,s.]
then A3: not p in ].s,t.[ by XBOOLE_0:def_5;
A4: r < p by A2, Th4;
( p <= s or t <= p ) by A3, Th4;
hence p in ].r,s.] by A2, A4, Th2, Th4; ::_thesis: verum
end;
assume A5: p in ].r,s.] ; ::_thesis: p in ].r,t.[ \ ].s,t.[
then A6: p <= s by Th2;
A7: r < p by A5, Th2;
p < t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.[ by A7, Th4;
not p in ].s,t.[ by A6, Th4;
hence p in ].r,t.[ \ ].s,t.[ by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:198
for p, q, r, s being ext-real number st [.p,q.[ meets [.r,s.[ holds
[.p,q.[ \ [.r,s.[ = [.p,r.[ \/ [.s,q.[
proof
let p, q, r, s be ext-real number ; ::_thesis: ( [.p,q.[ meets [.r,s.[ implies [.p,q.[ \ [.r,s.[ = [.p,r.[ \/ [.s,q.[ )
assume [.p,q.[ meets [.r,s.[ ; ::_thesis: [.p,q.[ \ [.r,s.[ = [.p,r.[ \/ [.s,q.[
then consider u being ext-real number such that
A1: u in [.r,s.[ and
A2: u in [.p,q.[ by MEMBERED:def_20;
A3: r <= u by A1, Th3;
A4: u <= s by A1, Th3;
A5: p <= u by A2, Th3;
u <= q by A2, Th3;
then A6: r <= q by A3, XXREAL_0:2;
A7: p <= s by A4, A5, XXREAL_0:2;
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in [.p,q.[ \ [.r,s.[ or t in [.p,r.[ \/ [.s,q.[ ) & ( not t in [.p,r.[ \/ [.s,q.[ or t in [.p,q.[ \ [.r,s.[ ) )
thus ( t in [.p,q.[ \ [.r,s.[ implies t in [.p,r.[ \/ [.s,q.[ ) ::_thesis: ( not t in [.p,r.[ \/ [.s,q.[ or t in [.p,q.[ \ [.r,s.[ )
proof
assume A8: t in [.p,q.[ \ [.r,s.[ ; ::_thesis: t in [.p,r.[ \/ [.s,q.[
then A9: not t in [.r,s.[ by XBOOLE_0:def_5;
A10: p <= t by A8, Th3;
A11: t < q by A8, Th3;
( t < r or s <= t ) by A9, Th3;
then ( t in [.p,r.[ or t in [.s,q.[ ) by A10, A11, Th3;
hence t in [.p,r.[ \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in [.p,r.[ \/ [.s,q.[ ; ::_thesis: t in [.p,q.[ \ [.r,s.[
then ( t in [.p,r.[ or t in [.s,q.[ ) by XBOOLE_0:def_3;
then A12: ( ( p <= t & t < r ) or ( s <= t & t < q ) ) by Th3;
then A13: p <= t by A7, XXREAL_0:2;
t < q by A6, A12, XXREAL_0:2;
then A14: t in [.p,q.[ by A13, Th3;
not t in [.r,s.[ by A12, Th3;
hence t in [.p,q.[ \ [.r,s.[ by A14, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:199
for p, q, r, s being ext-real number st ].p,q.] meets ].r,s.] holds
].p,q.] \ ].r,s.] = ].p,r.] \/ ].s,q.]
proof
let p, q, r, s be ext-real number ; ::_thesis: ( ].p,q.] meets ].r,s.] implies ].p,q.] \ ].r,s.] = ].p,r.] \/ ].s,q.] )
assume ].p,q.] meets ].r,s.] ; ::_thesis: ].p,q.] \ ].r,s.] = ].p,r.] \/ ].s,q.]
then consider u being ext-real number such that
A1: u in ].r,s.] and
A2: u in ].p,q.] by MEMBERED:def_20;
A3: r < u by A1, Th2;
A4: u <= s by A1, Th2;
A5: p < u by A2, Th2;
u <= q by A2, Th2;
then A6: r <= q by A3, XXREAL_0:2;
A7: p <= s by A4, A5, XXREAL_0:2;
let t be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not t in ].p,q.] \ ].r,s.] or t in ].p,r.] \/ ].s,q.] ) & ( not t in ].p,r.] \/ ].s,q.] or t in ].p,q.] \ ].r,s.] ) )
thus ( t in ].p,q.] \ ].r,s.] implies t in ].p,r.] \/ ].s,q.] ) ::_thesis: ( not t in ].p,r.] \/ ].s,q.] or t in ].p,q.] \ ].r,s.] )
proof
assume A8: t in ].p,q.] \ ].r,s.] ; ::_thesis: t in ].p,r.] \/ ].s,q.]
then A9: not t in ].r,s.] by XBOOLE_0:def_5;
A10: p < t by A8, Th2;
A11: t <= q by A8, Th2;
( t <= r or s < t ) by A9, Th2;
then ( t in ].p,r.] or t in ].s,q.] ) by A10, A11, Th2;
hence t in ].p,r.] \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume t in ].p,r.] \/ ].s,q.] ; ::_thesis: t in ].p,q.] \ ].r,s.]
then ( t in ].p,r.] or t in ].s,q.] ) by XBOOLE_0:def_3;
then A12: ( ( p < t & t <= r ) or ( s < t & t <= q ) ) by Th2;
then A13: p < t by A7, XXREAL_0:2;
t <= q by A6, A12, XXREAL_0:2;
then A14: t in ].p,q.] by A13, Th2;
not t in ].r,s.] by A12, Th2;
hence t in ].p,q.] \ ].r,s.] by A14, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:200
for p, r, s, q being ext-real number st p <= r & s <= q holds
[.p,q.] \ ([.p,r.] \/ [.s,q.]) = ].r,s.[
proof
let p, r, s, q be ext-real number ; ::_thesis: ( p <= r & s <= q implies [.p,q.] \ ([.p,r.] \/ [.s,q.]) = ].r,s.[ )
assume that
A1: p <= r and
A2: s <= q ; ::_thesis: [.p,q.] \ ([.p,r.] \/ [.s,q.]) = ].r,s.[
thus [.p,q.] \ ([.p,r.] \/ [.s,q.]) = ([.p,q.] \ [.p,r.]) \ [.s,q.] by XBOOLE_1:41
.= ].r,q.] \ [.s,q.] by A1, Th182
.= ].r,s.[ by A2, Th191 ; ::_thesis: verum
end;
theorem :: XXREAL_1:201
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,t.] \ {s} = [.r,s.[ \/ ].s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,t.] \ {s} = [.r,s.[ \/ ].s,t.] )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,t.] \ {s} = [.r,s.[ \/ ].s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.] \ {s} or p in [.r,s.[ \/ ].s,t.] ) & ( not p in [.r,s.[ \/ ].s,t.] or p in [.r,t.] \ {s} ) )
thus ( p in [.r,t.] \ {s} implies p in [.r,s.[ \/ ].s,t.] ) ::_thesis: ( not p in [.r,s.[ \/ ].s,t.] or p in [.r,t.] \ {s} )
proof
assume A3: p in [.r,t.] \ {s} ; ::_thesis: p in [.r,s.[ \/ ].s,t.]
then not p in {s} by XBOOLE_0:def_5;
then p <> s by TARSKI:def_1;
then ( ( r <= p & p < s ) or ( s < p & p <= t ) ) by A3, Th1, XXREAL_0:1;
then ( p in [.r,s.[ or p in ].s,t.] ) by Th2, Th3;
hence p in [.r,s.[ \/ ].s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume p in [.r,s.[ \/ ].s,t.] ; ::_thesis: p in [.r,t.] \ {s}
then ( p in [.r,s.[ or p in ].s,t.] ) by XBOOLE_0:def_3;
then A4: ( ( r <= p & p < s ) or ( s < p & p <= t ) ) by Th2, Th3;
then A5: r <= p by A1, XXREAL_0:2;
p <= t by A2, A4, XXREAL_0:2;
then A6: p in [.r,t.] by A5, Th1;
not p in {s} by A4, TARSKI:def_1;
hence p in [.r,t.] \ {s} by A6, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:202
for r, s, t being ext-real number st r <= s & s < t holds
[.r,t.[ \ {s} = [.r,s.[ \/ ].s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s < t implies [.r,t.[ \ {s} = [.r,s.[ \/ ].s,t.[ )
assume that
A1: r <= s and
A2: s < t ; ::_thesis: [.r,t.[ \ {s} = [.r,s.[ \/ ].s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,t.[ \ {s} or p in [.r,s.[ \/ ].s,t.[ ) & ( not p in [.r,s.[ \/ ].s,t.[ or p in [.r,t.[ \ {s} ) )
thus ( p in [.r,t.[ \ {s} implies p in [.r,s.[ \/ ].s,t.[ ) ::_thesis: ( not p in [.r,s.[ \/ ].s,t.[ or p in [.r,t.[ \ {s} )
proof
assume A3: p in [.r,t.[ \ {s} ; ::_thesis: p in [.r,s.[ \/ ].s,t.[
then not p in {s} by XBOOLE_0:def_5;
then p <> s by TARSKI:def_1;
then ( ( r <= p & p < s ) or ( s < p & p < t ) ) by A3, Th3, XXREAL_0:1;
then ( p in [.r,s.[ or p in ].s,t.[ ) by Th3, Th4;
hence p in [.r,s.[ \/ ].s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume p in [.r,s.[ \/ ].s,t.[ ; ::_thesis: p in [.r,t.[ \ {s}
then ( p in [.r,s.[ or p in ].s,t.[ ) by XBOOLE_0:def_3;
then A4: ( ( r <= p & p < s ) or ( s < p & p < t ) ) by Th3, Th4;
then A5: r <= p by A1, XXREAL_0:2;
p < t by A2, A4, XXREAL_0:2;
then A6: p in [.r,t.[ by A5, Th3;
not p in {s} by A4, TARSKI:def_1;
hence p in [.r,t.[ \ {s} by A6, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:203
for r, s, t being ext-real number st r < s & s <= t holds
].r,t.] \ {s} = ].r,s.[ \/ ].s,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s <= t implies ].r,t.] \ {s} = ].r,s.[ \/ ].s,t.] )
assume that
A1: r < s and
A2: s <= t ; ::_thesis: ].r,t.] \ {s} = ].r,s.[ \/ ].s,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.] \ {s} or p in ].r,s.[ \/ ].s,t.] ) & ( not p in ].r,s.[ \/ ].s,t.] or p in ].r,t.] \ {s} ) )
thus ( p in ].r,t.] \ {s} implies p in ].r,s.[ \/ ].s,t.] ) ::_thesis: ( not p in ].r,s.[ \/ ].s,t.] or p in ].r,t.] \ {s} )
proof
assume A3: p in ].r,t.] \ {s} ; ::_thesis: p in ].r,s.[ \/ ].s,t.]
then not p in {s} by XBOOLE_0:def_5;
then p <> s by TARSKI:def_1;
then ( ( r < p & p < s ) or ( s < p & p <= t ) ) by A3, Th2, XXREAL_0:1;
then ( p in ].r,s.[ or p in ].s,t.] ) by Th2, Th4;
hence p in ].r,s.[ \/ ].s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume p in ].r,s.[ \/ ].s,t.] ; ::_thesis: p in ].r,t.] \ {s}
then ( p in ].r,s.[ or p in ].s,t.] ) by XBOOLE_0:def_3;
then A4: ( ( r < p & p < s ) or ( s < p & p <= t ) ) by Th2, Th4;
then A5: r < p by A1, XXREAL_0:2;
p <= t by A2, A4, XXREAL_0:2;
then A6: p in ].r,t.] by A5, Th2;
not p in {s} by A4, TARSKI:def_1;
hence p in ].r,t.] \ {s} by A6, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:204
for r, s, t being ext-real number st r < s & s < t holds
].r,t.[ \ {s} = ].r,s.[ \/ ].s,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s < t implies ].r,t.[ \ {s} = ].r,s.[ \/ ].s,t.[ )
assume that
A1: r < s and
A2: s < t ; ::_thesis: ].r,t.[ \ {s} = ].r,s.[ \/ ].s,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,t.[ \ {s} or p in ].r,s.[ \/ ].s,t.[ ) & ( not p in ].r,s.[ \/ ].s,t.[ or p in ].r,t.[ \ {s} ) )
thus ( p in ].r,t.[ \ {s} implies p in ].r,s.[ \/ ].s,t.[ ) ::_thesis: ( not p in ].r,s.[ \/ ].s,t.[ or p in ].r,t.[ \ {s} )
proof
assume A3: p in ].r,t.[ \ {s} ; ::_thesis: p in ].r,s.[ \/ ].s,t.[
then not p in {s} by XBOOLE_0:def_5;
then p <> s by TARSKI:def_1;
then ( ( r < p & p < s ) or ( s < p & p < t ) ) by A3, Th4, XXREAL_0:1;
then ( p in ].r,s.[ or p in ].s,t.[ ) by Th4;
hence p in ].r,s.[ \/ ].s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume p in ].r,s.[ \/ ].s,t.[ ; ::_thesis: p in ].r,t.[ \ {s}
then ( p in ].r,s.[ or p in ].s,t.[ ) by XBOOLE_0:def_3;
then A4: ( ( r < p & p < s ) or ( s < p & p < t ) ) by Th4;
then A5: r < p by A1, XXREAL_0:2;
p < t by A2, A4, XXREAL_0:2;
then A6: p in ].r,t.[ by A5, Th4;
not p in {s} by A4, TARSKI:def_1;
hence p in ].r,t.[ \ {s} by A6, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem :: XXREAL_1:205
for s, r, t being ext-real number holds not s in ].r,s.[ \/ ].s,t.[
proof
let s, r, t be ext-real number ; ::_thesis: not s in ].r,s.[ \/ ].s,t.[
assume s in ].r,s.[ \/ ].s,t.[ ; ::_thesis: contradiction
then ( s in ].r,s.[ or s in ].s,t.[ ) by XBOOLE_0:def_3;
hence contradiction by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:206
for s, r, t being ext-real number holds not s in [.r,s.[ \/ ].s,t.[
proof
let s, r, t be ext-real number ; ::_thesis: not s in [.r,s.[ \/ ].s,t.[
assume s in [.r,s.[ \/ ].s,t.[ ; ::_thesis: contradiction
then ( s in [.r,s.[ or s in ].s,t.[ ) by XBOOLE_0:def_3;
hence contradiction by Th3, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:207
for s, r, t being ext-real number holds not s in ].r,s.[ \/ ].s,t.]
proof
let s, r, t be ext-real number ; ::_thesis: not s in ].r,s.[ \/ ].s,t.]
assume s in ].r,s.[ \/ ].s,t.] ; ::_thesis: contradiction
then ( s in ].r,s.[ or s in ].s,t.] ) by XBOOLE_0:def_3;
hence contradiction by Th2, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:208
for s, r, t being ext-real number holds not s in [.r,s.[ \/ ].s,t.]
proof
let s, r, t be ext-real number ; ::_thesis: not s in [.r,s.[ \/ ].s,t.]
assume s in [.r,s.[ \/ ].s,t.] ; ::_thesis: contradiction
then ( s in [.r,s.[ or s in ].s,t.] ) by XBOOLE_0:def_3;
hence contradiction by Th2, Th3; ::_thesis: verum
end;
begin
theorem :: XXREAL_1:209
[.-infty,+infty.] = ExtREAL
proof
let r be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not r in [.-infty,+infty.] or r in ExtREAL ) & ( not r in ExtREAL or r in [.-infty,+infty.] ) )
thus ( r in [.-infty,+infty.] implies r in ExtREAL ) by XXREAL_0:def_1; ::_thesis: ( not r in ExtREAL or r in [.-infty,+infty.] )
assume r in ExtREAL ; ::_thesis: r in [.-infty,+infty.]
A1: -infty <= r by XXREAL_0:5;
r <= +infty by XXREAL_0:3;
hence r in [.-infty,+infty.] by A1, Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:210
for p being ext-real number holds ].p,-infty.[ = {}
proof
let p be ext-real number ; ::_thesis: ].p,-infty.[ = {}
for x being set holds not x in ].p,-infty.[
proof
given x being set such that A1: x in ].p,-infty.[ ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
s < -infty by A1, Th4;
hence contradiction by XXREAL_0:5; ::_thesis: verum
end;
hence ].p,-infty.[ = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:211
for p being ext-real number holds [.p,-infty.[ = {}
proof
let p be ext-real number ; ::_thesis: [.p,-infty.[ = {}
for x being set holds not x in [.p,-infty.[
proof
given x being set such that A1: x in [.p,-infty.[ ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
s < -infty by A1, Th3;
hence contradiction by XXREAL_0:5; ::_thesis: verum
end;
hence [.p,-infty.[ = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:212
for p being ext-real number holds ].p,-infty.] = {}
proof
let p be ext-real number ; ::_thesis: ].p,-infty.] = {}
for x being set holds not x in ].p,-infty.]
proof
given x being set such that A1: x in ].p,-infty.] ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
A2: p < s by A1, Th2;
s <= -infty by A1, Th2;
then p < -infty by A2, XXREAL_0:2;
hence contradiction by XXREAL_0:5; ::_thesis: verum
end;
hence ].p,-infty.] = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:213
for p being ext-real number st p <> -infty holds
[.p,-infty.] = {}
proof
let p be ext-real number ; ::_thesis: ( p <> -infty implies [.p,-infty.] = {} )
assume A1: p <> -infty ; ::_thesis: [.p,-infty.] = {}
for x being set holds not x in [.p,-infty.]
proof
given x being set such that A2: x in [.p,-infty.] ; ::_thesis: contradiction
reconsider s = x as ext-real number by A2;
A3: p <= s by A2, Th1;
s <= -infty by A2, Th1;
hence contradiction by A1, A3, XXREAL_0:2, XXREAL_0:6; ::_thesis: verum
end;
hence [.p,-infty.] = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:214
for p being ext-real number holds ].+infty,p.[ = {}
proof
let p be ext-real number ; ::_thesis: ].+infty,p.[ = {}
for x being set holds not x in ].+infty,p.[
proof
given x being set such that A1: x in ].+infty,p.[ ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
+infty < s by A1, Th4;
hence contradiction by XXREAL_0:3; ::_thesis: verum
end;
hence ].+infty,p.[ = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:215
for p being ext-real number holds [.+infty,p.[ = {}
proof
let p be ext-real number ; ::_thesis: [.+infty,p.[ = {}
for x being set holds not x in [.+infty,p.[
proof
given x being set such that A1: x in [.+infty,p.[ ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
A2: +infty <= s by A1, Th3;
s < p by A1, Th3;
then p > +infty by A2, XXREAL_0:2;
hence contradiction by XXREAL_0:3; ::_thesis: verum
end;
hence [.+infty,p.[ = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:216
for p being ext-real number holds ].+infty,p.] = {}
proof
let p be ext-real number ; ::_thesis: ].+infty,p.] = {}
for x being set holds not x in ].+infty,p.]
proof
given x being set such that A1: x in ].+infty,p.] ; ::_thesis: contradiction
reconsider s = x as ext-real number by A1;
+infty < s by A1, Th2;
hence contradiction by XXREAL_0:3; ::_thesis: verum
end;
hence ].+infty,p.] = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:217
for p being ext-real number st p <> +infty holds
[.+infty,p.] = {}
proof
let p be ext-real number ; ::_thesis: ( p <> +infty implies [.+infty,p.] = {} )
assume A1: p <> +infty ; ::_thesis: [.+infty,p.] = {}
for x being set holds not x in [.+infty,p.]
proof
given x being set such that A2: x in [.+infty,p.] ; ::_thesis: contradiction
reconsider s = x as ext-real number by A2;
A3: +infty <= s by A2, Th1;
s <= p by A2, Th1;
hence contradiction by A1, A3, XXREAL_0:2, XXREAL_0:4; ::_thesis: verum
end;
hence [.+infty,p.] = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:218
for p, q being ext-real number st p > q holds
p in ].q,+infty.]
proof
let p, q be ext-real number ; ::_thesis: ( p > q implies p in ].q,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( p > q implies p in ].q,+infty.] ) by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:219
for q, p being ext-real number st q <= p holds
p in [.q,+infty.]
proof
let q, p be ext-real number ; ::_thesis: ( q <= p implies p in [.q,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( q <= p implies p in [.q,+infty.] ) by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:220
for p, q being ext-real number st p <= q holds
p in [.-infty,q.]
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies p in [.-infty,q.] )
p >= -infty by XXREAL_0:5;
hence ( p <= q implies p in [.-infty,q.] ) by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:221
for p, q being ext-real number st p < q holds
p in [.-infty,q.[
proof
let p, q be ext-real number ; ::_thesis: ( p < q implies p in [.-infty,q.[ )
p >= -infty by XXREAL_0:5;
hence ( p < q implies p in [.-infty,q.[ ) by Th3; ::_thesis: verum
end;
begin
theorem :: XXREAL_1:222
for p, q being ext-real number st p <= q holds
[.p,q.] = [.p,q.] \/ [.q,p.]
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.p,q.] = [.p,q.] \/ [.q,p.] )
assume A1: p <= q ; ::_thesis: [.p,q.] = [.p,q.] \/ [.q,p.]
then A2: [.q,p.] c= {p} by Th85;
p in [.p,q.] by A1, Th1;
then {p} c= [.p,q.] by ZFMISC_1:31;
hence [.p,q.] = [.p,q.] \/ [.q,p.] by A2, XBOOLE_1:1, XBOOLE_1:12; ::_thesis: verum
end;
theorem :: XXREAL_1:223
for r, s, t, p being ext-real number st r <= s & s <= t holds
not r in ].s,t.[ \/ ].t,p.[
proof
let r, s, t, p be ext-real number ; ::_thesis: ( r <= s & s <= t implies not r in ].s,t.[ \/ ].t,p.[ )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: not r in ].s,t.[ \/ ].t,p.[
assume r in ].s,t.[ \/ ].t,p.[ ; ::_thesis: contradiction
then ( r in ].s,t.[ or r in ].t,p.[ ) by XBOOLE_0:def_3;
then ( ( s < r & r < t ) or ( t < r & r < p ) ) by Th4;
hence contradiction by A1, A2, XXREAL_0:2; ::_thesis: verum
end;
theorem Th224: :: XXREAL_1:224
REAL = ].-infty,+infty.[
proof
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in REAL or x in ].-infty,+infty.[ ) & ( not x in ].-infty,+infty.[ or x in REAL ) )
thus ( x in REAL implies x in ].-infty,+infty.[ ) ::_thesis: ( not x in ].-infty,+infty.[ or x in REAL )
proof
assume A1: x in REAL ; ::_thesis: x in ].-infty,+infty.[
then A2: -infty < x by XXREAL_0:12;
x < +infty by A1, XXREAL_0:9;
hence x in ].-infty,+infty.[ by A2, Th4; ::_thesis: verum
end;
assume A3: x in ].-infty,+infty.[ ; ::_thesis: x in REAL
then A4: -infty < x by Th4;
x < +infty by A3, Th4;
hence x in REAL by A4, XXREAL_0:14; ::_thesis: verum
end;
theorem Th225: :: XXREAL_1:225
for p, q being ext-real number holds ].p,q.[ c= REAL
proof
let p, q be ext-real number ; ::_thesis: ].p,q.[ c= REAL
let x be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not x in ].p,q.[ or x in REAL )
assume A1: x in ].p,q.[ ; ::_thesis: x in REAL
then A2: p < x by Th4;
x < q by A1, Th4;
hence x in REAL by A2, XXREAL_0:48; ::_thesis: verum
end;
theorem Th226: :: XXREAL_1:226
for p, q being ext-real number st p in REAL holds
[.p,q.[ c= REAL
proof
let p, q be ext-real number ; ::_thesis: ( p in REAL implies [.p,q.[ c= REAL )
assume A1: p in REAL ; ::_thesis: [.p,q.[ c= REAL
let x be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not x in [.p,q.[ or x in REAL )
assume A2: x in [.p,q.[ ; ::_thesis: x in REAL
then A3: p <= x by Th3;
x < q by A2, Th3;
hence x in REAL by A1, A3, XXREAL_0:46; ::_thesis: verum
end;
theorem Th227: :: XXREAL_1:227
for q, p being ext-real number st q in REAL holds
].p,q.] c= REAL
proof
let q, p be ext-real number ; ::_thesis: ( q in REAL implies ].p,q.] c= REAL )
assume A1: q in REAL ; ::_thesis: ].p,q.] c= REAL
let x be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not x in ].p,q.] or x in REAL )
assume A2: x in ].p,q.] ; ::_thesis: x in REAL
then A3: p < x by Th2;
x <= q by A2, Th2;
hence x in REAL by A1, A3, XXREAL_0:47; ::_thesis: verum
end;
theorem Th228: :: XXREAL_1:228
for p, q being ext-real number st p in REAL & q in REAL holds
[.p,q.] c= REAL
proof
let p, q be ext-real number ; ::_thesis: ( p in REAL & q in REAL implies [.p,q.] c= REAL )
assume that
A1: p in REAL and
A2: q in REAL ; ::_thesis: [.p,q.] c= REAL
let x be ext-real number ; :: according to MEMBERED:def_8 ::_thesis: ( not x in [.p,q.] or x in REAL )
assume A3: x in [.p,q.] ; ::_thesis: x in REAL
then A4: p <= x by Th1;
x <= q by A3, Th1;
hence x in REAL by A1, A2, A4, XXREAL_0:45; ::_thesis: verum
end;
registration
let p, q be ext-real number ;
cluster].p,q.[ -> real-membered ;
coherence
].p,q.[ is real-membered by Th225, MEMBERED:21;
end;
registration
let p be real number ;
let q be ext-real number ;
cluster[.p,q.[ -> real-membered ;
coherence
[.p,q.[ is real-membered
proof
p in REAL by XREAL_0:def_1;
then [.p,q.[ c= REAL by Th226;
hence [.p,q.[ is real-membered ; ::_thesis: verum
end;
end;
registration
let q be real number ;
let p be ext-real number ;
cluster].p,q.] -> real-membered ;
coherence
].p,q.] is real-membered
proof
q in REAL by XREAL_0:def_1;
then ].p,q.] c= REAL by Th227;
hence ].p,q.] is real-membered ; ::_thesis: verum
end;
end;
registration
let p, q be real number ;
cluster[.p,q.] -> real-membered ;
coherence
[.p,q.] is real-membered
proof
A1: p in REAL by XREAL_0:def_1;
q in REAL by XREAL_0:def_1;
then [.p,q.] c= REAL by A1, Th228;
hence [.p,q.] is real-membered ; ::_thesis: verum
end;
end;
theorem :: XXREAL_1:229
for s being ext-real number holds ].-infty,s.[ = { g where g is Real : g < s }
proof
let s be ext-real number ; ::_thesis: ].-infty,s.[ = { g where g is Real : g < s }
thus ].-infty,s.[ c= { g where g is Real : g < s } :: according to XBOOLE_0:def_10 ::_thesis: { g where g is Real : g < s } c= ].-infty,s.[
proof
let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in ].-infty,s.[ or x in { g where g is Real : g < s } )
assume A1: x in ].-infty,s.[ ; ::_thesis: x in { g where g is Real : g < s }
then A2: -infty < x by Th4;
A3: x < s by A1, Th4;
then x in REAL by A2, XXREAL_0:48;
hence x in { g where g is Real : g < s } by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { g where g is Real : g < s } or x in ].-infty,s.[ )
assume x in { g where g is Real : g < s } ; ::_thesis: x in ].-infty,s.[
then consider g being Real such that
A4: x = g and
A5: g < s ;
-infty < g by XXREAL_0:12;
hence x in ].-infty,s.[ by A4, A5, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:230
for s being ext-real number holds ].s,+infty.[ = { g where g is Real : s < g }
proof
let s be ext-real number ; ::_thesis: ].s,+infty.[ = { g where g is Real : s < g }
thus ].s,+infty.[ c= { g where g is Real : s < g } :: according to XBOOLE_0:def_10 ::_thesis: { g where g is Real : s < g } c= ].s,+infty.[
proof
let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in ].s,+infty.[ or x in { g where g is Real : s < g } )
assume A1: x in ].s,+infty.[ ; ::_thesis: x in { g where g is Real : s < g }
then A2: s < x by Th4;
x < +infty by A1, Th4;
then x in REAL by A2, XXREAL_0:48;
hence x in { g where g is Real : s < g } by A2; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { g where g is Real : s < g } or x in ].s,+infty.[ )
assume x in { g where g is Real : s < g } ; ::_thesis: x in ].s,+infty.[
then consider g being Real such that
A3: x = g and
A4: s < g ;
g < +infty by XXREAL_0:9;
hence x in ].s,+infty.[ by A3, A4, Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:231
for s being real number holds ].-infty,s.] = { g where g is Real : g <= s }
proof
let s be real number ; ::_thesis: ].-infty,s.] = { g where g is Real : g <= s }
thus ].-infty,s.] c= { g where g is Real : g <= s } :: according to XBOOLE_0:def_10 ::_thesis: { g where g is Real : g <= s } c= ].-infty,s.]
proof
let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in ].-infty,s.] or x in { g where g is Real : g <= s } )
assume x in ].-infty,s.] ; ::_thesis: x in { g where g is Real : g <= s }
then A1: x <= s by Th2;
x in REAL by XREAL_0:def_1;
hence x in { g where g is Real : g <= s } by A1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { g where g is Real : g <= s } or x in ].-infty,s.] )
assume x in { g where g is Real : g <= s } ; ::_thesis: x in ].-infty,s.]
then consider g being Real such that
A2: x = g and
A3: g <= s ;
-infty < g by XXREAL_0:12;
hence x in ].-infty,s.] by A2, A3, Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:232
for s being real number holds [.s,+infty.[ = { g where g is Real : s <= g }
proof
let s be real number ; ::_thesis: [.s,+infty.[ = { g where g is Real : s <= g }
thus [.s,+infty.[ c= { g where g is Real : s <= g } :: according to XBOOLE_0:def_10 ::_thesis: { g where g is Real : s <= g } c= [.s,+infty.[
proof
let x be real number ; :: according to MEMBERED:def_9 ::_thesis: ( not x in [.s,+infty.[ or x in { g where g is Real : s <= g } )
assume x in [.s,+infty.[ ; ::_thesis: x in { g where g is Real : s <= g }
then A1: s <= x by Th3;
x in REAL by XREAL_0:def_1;
hence x in { g where g is Real : s <= g } by A1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { g where g is Real : s <= g } or x in [.s,+infty.[ )
assume x in { g where g is Real : s <= g } ; ::_thesis: x in [.s,+infty.[
then consider g being Real such that
A2: x = g and
A3: s <= g ;
g < +infty by XXREAL_0:9;
hence x in [.s,+infty.[ by A2, A3, Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:233
for u being ext-real number
for x being real number holds
( x in ].-infty,u.[ iff x < u )
proof
let u be ext-real number ; ::_thesis: for x being real number holds
( x in ].-infty,u.[ iff x < u )
let x be real number ; ::_thesis: ( x in ].-infty,u.[ iff x < u )
x in REAL by XREAL_0:def_1;
then -infty < x by XXREAL_0:12;
hence ( x in ].-infty,u.[ iff x < u ) by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:234
for u being ext-real number
for x being real number holds
( x in ].-infty,u.] iff x <= u )
proof
let u be ext-real number ; ::_thesis: for x being real number holds
( x in ].-infty,u.] iff x <= u )
let x be real number ; ::_thesis: ( x in ].-infty,u.] iff x <= u )
x in REAL by XREAL_0:def_1;
then -infty < x by XXREAL_0:12;
hence ( x in ].-infty,u.] iff x <= u ) by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:235
for u being ext-real number
for x being real number holds
( x in ].u,+infty.[ iff u < x )
proof
let u be ext-real number ; ::_thesis: for x being real number holds
( x in ].u,+infty.[ iff u < x )
let x be real number ; ::_thesis: ( x in ].u,+infty.[ iff u < x )
x in REAL by XREAL_0:def_1;
then x < +infty by XXREAL_0:9;
hence ( x in ].u,+infty.[ iff u < x ) by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:236
for u being ext-real number
for x being real number holds
( x in [.u,+infty.[ iff u <= x )
proof
let u be ext-real number ; ::_thesis: for x being real number holds
( x in [.u,+infty.[ iff u <= x )
let x be real number ; ::_thesis: ( x in [.u,+infty.[ iff u <= x )
x in REAL by XREAL_0:def_1;
then x < +infty by XXREAL_0:9;
hence ( x in [.u,+infty.[ iff u <= x ) by Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:237
for p, r, s being ext-real number st p <= r holds
[.r,s.] c= [.p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies [.r,s.] c= [.p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies [.r,s.] c= [.p,+infty.] ) by Th34; ::_thesis: verum
end;
theorem :: XXREAL_1:238
for p, r, s being ext-real number st p <= r holds
[.r,s.[ c= [.p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies [.r,s.[ c= [.p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies [.r,s.[ c= [.p,+infty.] ) by Th35; ::_thesis: verum
end;
theorem :: XXREAL_1:239
for p, r, s being ext-real number st p <= r holds
].r,s.] c= [.p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.] c= [.p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.] c= [.p,+infty.] ) by Th36; ::_thesis: verum
end;
theorem :: XXREAL_1:240
for p, r, s being ext-real number st p <= r holds
].r,s.[ c= [.p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.[ c= [.p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.[ c= [.p,+infty.] ) by Th37; ::_thesis: verum
end;
theorem :: XXREAL_1:241
for p, r, s being ext-real number st p <= r holds
[.r,s.[ c= [.p,+infty.[
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies [.r,s.[ c= [.p,+infty.[ )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies [.r,s.[ c= [.p,+infty.[ ) by Th38; ::_thesis: verum
end;
theorem :: XXREAL_1:242
for p, r, s being ext-real number st p < r holds
[.r,s.] c= ].p,+infty.] by Th39, XXREAL_0:3;
theorem :: XXREAL_1:243
for p, r, s being ext-real number st p < r holds
[.r,s.[ c= ].p,+infty.] by Th40, XXREAL_0:3;
theorem :: XXREAL_1:244
for p, r, s being ext-real number st p <= r holds
].r,s.[ c= ].p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.[ c= ].p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.[ c= ].p,+infty.] ) by Th41; ::_thesis: verum
end;
theorem :: XXREAL_1:245
for p, r, s being ext-real number st p <= r holds
].r,s.] c= ].p,+infty.]
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.] c= ].p,+infty.] )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.] c= ].p,+infty.] ) by Th42; ::_thesis: verum
end;
theorem :: XXREAL_1:246
for p, r, s being ext-real number st p <= r holds
].r,s.[ c= [.p,+infty.[
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.[ c= [.p,+infty.[ )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.[ c= [.p,+infty.[ ) by Th45; ::_thesis: verum
end;
theorem :: XXREAL_1:247
for p, r, s being ext-real number st p <= r holds
].r,s.[ c= ].p,+infty.[
proof
let p, r, s be ext-real number ; ::_thesis: ( p <= r implies ].r,s.[ c= ].p,+infty.[ )
s <= +infty by XXREAL_0:3;
hence ( p <= r implies ].r,s.[ c= ].p,+infty.[ ) by Th46; ::_thesis: verum
end;
theorem :: XXREAL_1:248
for p, r, s being ext-real number st p < r holds
[.r,s.[ c= ].p,+infty.[ by Th48, XXREAL_0:3;
theorem :: XXREAL_1:249
for p, r being ext-real number
for s being real number st p < r holds
[.r,s.] c= ].p,+infty.[
proof
let p, r be ext-real number ; ::_thesis: for s being real number st p < r holds
[.r,s.] c= ].p,+infty.[
let s be real number ; ::_thesis: ( p < r implies [.r,s.] c= ].p,+infty.[ )
s in REAL by XREAL_0:def_1;
then s < +infty by XXREAL_0:9;
hence ( p < r implies [.r,s.] c= ].p,+infty.[ ) by Th47; ::_thesis: verum
end;
theorem :: XXREAL_1:250
for p, r being ext-real number
for s being real number st p <= r holds
].r,s.] c= ].p,+infty.[
proof
let p, r be ext-real number ; ::_thesis: for s being real number st p <= r holds
].r,s.] c= ].p,+infty.[
let s be real number ; ::_thesis: ( p <= r implies ].r,s.] c= ].p,+infty.[ )
s in REAL by XREAL_0:def_1;
hence ( p <= r implies ].r,s.] c= ].p,+infty.[ ) by Th49, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:251
for p, r being ext-real number
for s being real number st p <= r holds
[.r,s.] c= [.p,+infty.[
proof
let p, r be ext-real number ; ::_thesis: for s being real number st p <= r holds
[.r,s.] c= [.p,+infty.[
let s be real number ; ::_thesis: ( p <= r implies [.r,s.] c= [.p,+infty.[ )
s in REAL by XREAL_0:def_1;
hence ( p <= r implies [.r,s.] c= [.p,+infty.[ ) by Th43, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:252
for p, r being ext-real number
for s being real number st p <= r holds
].r,s.] c= [.p,+infty.[
proof
let p, r be ext-real number ; ::_thesis: for s being real number st p <= r holds
].r,s.] c= [.p,+infty.[
let s be real number ; ::_thesis: ( p <= r implies ].r,s.] c= [.p,+infty.[ )
s in REAL by XREAL_0:def_1;
hence ( p <= r implies ].r,s.] c= [.p,+infty.[ ) by Th44, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:253
for s, q, r being ext-real number st s <= q holds
[.r,s.] c= [.-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies [.r,s.] c= [.-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies [.r,s.] c= [.-infty,q.] ) by Th34; ::_thesis: verum
end;
theorem :: XXREAL_1:254
for s, q, r being ext-real number st s <= q holds
[.r,s.[ c= [.-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies [.r,s.[ c= [.-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies [.r,s.[ c= [.-infty,q.] ) by Th35; ::_thesis: verum
end;
theorem :: XXREAL_1:255
for s, q, r being ext-real number st s <= q holds
].r,s.] c= [.-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.] c= [.-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.] c= [.-infty,q.] ) by Th36; ::_thesis: verum
end;
theorem :: XXREAL_1:256
for s, q, r being ext-real number st s <= q holds
].r,s.[ c= [.-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.[ c= [.-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.[ c= [.-infty,q.] ) by Th37; ::_thesis: verum
end;
theorem :: XXREAL_1:257
for s, q, r being ext-real number st s <= q holds
[.r,s.[ c= [.-infty,q.[
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies [.r,s.[ c= [.-infty,q.[ )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies [.r,s.[ c= [.-infty,q.[ ) by Th38; ::_thesis: verum
end;
theorem :: XXREAL_1:258
for s, q, r being ext-real number st s <= q holds
].r,s.[ c= ].-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.[ c= ].-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.[ c= ].-infty,q.] ) by Th41; ::_thesis: verum
end;
theorem :: XXREAL_1:259
for s, q, r being ext-real number st s <= q holds
].r,s.] c= ].-infty,q.]
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.] c= ].-infty,q.] )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.] c= ].-infty,q.] ) by Th42; ::_thesis: verum
end;
theorem :: XXREAL_1:260
for s, q, r being ext-real number st s < q holds
[.r,s.] c= [.-infty,q.[ by Th43, XXREAL_0:5;
theorem :: XXREAL_1:261
for s, q, r being ext-real number st s < q holds
].r,s.] c= [.-infty,q.[ by Th44, XXREAL_0:5;
theorem :: XXREAL_1:262
for s, q, r being ext-real number st s <= q holds
].r,s.[ c= [.-infty,q.[
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.[ c= [.-infty,q.[ )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.[ c= [.-infty,q.[ ) by Th45; ::_thesis: verum
end;
theorem :: XXREAL_1:263
for s, q, r being ext-real number st s <= q holds
].r,s.[ c= ].-infty,q.[
proof
let s, q, r be ext-real number ; ::_thesis: ( s <= q implies ].r,s.[ c= ].-infty,q.[ )
-infty <= r by XXREAL_0:5;
hence ( s <= q implies ].r,s.[ c= ].-infty,q.[ ) by Th46; ::_thesis: verum
end;
theorem :: XXREAL_1:264
for s, q, r being ext-real number st s < q holds
].r,s.] c= ].-infty,q.[ by Th49, XXREAL_0:5;
theorem :: XXREAL_1:265
for s, q being ext-real number
for r being real number st s <= q holds
[.r,s.] c= ].-infty,q.]
proof
let s, q be ext-real number ; ::_thesis: for r being real number st s <= q holds
[.r,s.] c= ].-infty,q.]
let r be real number ; ::_thesis: ( s <= q implies [.r,s.] c= ].-infty,q.] )
r in REAL by XREAL_0:def_1;
hence ( s <= q implies [.r,s.] c= ].-infty,q.] ) by Th39, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:266
for s, q being ext-real number
for r being real number st s <= q holds
[.r,s.[ c= ].-infty,q.]
proof
let s, q be ext-real number ; ::_thesis: for r being real number st s <= q holds
[.r,s.[ c= ].-infty,q.]
let r be real number ; ::_thesis: ( s <= q implies [.r,s.[ c= ].-infty,q.] )
r in REAL by XREAL_0:def_1;
hence ( s <= q implies [.r,s.[ c= ].-infty,q.] ) by Th40, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:267
for s, q being ext-real number
for r being real number st s < q holds
[.r,s.] c= ].-infty,q.[
proof
let s, q be ext-real number ; ::_thesis: for r being real number st s < q holds
[.r,s.] c= ].-infty,q.[
let r be real number ; ::_thesis: ( s < q implies [.r,s.] c= ].-infty,q.[ )
r in REAL by XREAL_0:def_1;
then -infty < r by XXREAL_0:12;
hence ( s < q implies [.r,s.] c= ].-infty,q.[ ) by Th47; ::_thesis: verum
end;
theorem :: XXREAL_1:268
for s, q being ext-real number
for r being real number st s <= q holds
[.r,s.[ c= ].-infty,q.[
proof
let s, q be ext-real number ; ::_thesis: for r being real number st s <= q holds
[.r,s.[ c= ].-infty,q.[
let r be real number ; ::_thesis: ( s <= q implies [.r,s.[ c= ].-infty,q.[ )
r in REAL by XREAL_0:def_1;
hence ( s <= q implies [.r,s.[ c= ].-infty,q.[ ) by Th48, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:269
for a, b being ext-real number holds ].-infty,b.[ /\ ].a,+infty.[ = ].a,b.[
proof
let a, b be ext-real number ; ::_thesis: ].-infty,b.[ /\ ].a,+infty.[ = ].a,b.[
A1: -infty <= a by XXREAL_0:5;
b <= +infty by XXREAL_0:3;
hence ].-infty,b.[ /\ ].a,+infty.[ = ].a,b.[ by A1, Th160; ::_thesis: verum
end;
theorem :: XXREAL_1:270
for p being ext-real number
for b being real number holds ].-infty,b.] /\ ].p,+infty.[ = ].p,b.]
proof
let p be ext-real number ; ::_thesis: for b being real number holds ].-infty,b.] /\ ].p,+infty.[ = ].p,b.]
let b be real number ; ::_thesis: ].-infty,b.] /\ ].p,+infty.[ = ].p,b.]
A1: b in REAL by XREAL_0:def_1;
-infty <= p by XXREAL_0:5;
hence ].-infty,b.] /\ ].p,+infty.[ = ].p,b.] by A1, Th159, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:271
for p being ext-real number
for a being real number holds ].-infty,p.[ /\ [.a,+infty.[ = [.a,p.[
proof
let p be ext-real number ; ::_thesis: for a being real number holds ].-infty,p.[ /\ [.a,+infty.[ = [.a,p.[
let a be real number ; ::_thesis: ].-infty,p.[ /\ [.a,+infty.[ = [.a,p.[
a in REAL by XREAL_0:def_1;
then -infty < a by XXREAL_0:12;
hence ].-infty,p.[ /\ [.a,+infty.[ = [.a,p.[ by Th154, XXREAL_0:3; ::_thesis: verum
end;
theorem :: XXREAL_1:272
for a, b being real number holds ].-infty,a.] /\ [.b,+infty.[ = [.b,a.]
proof
let a, b be real number ; ::_thesis: ].-infty,a.] /\ [.b,+infty.[ = [.b,a.]
A1: a in REAL by XREAL_0:def_1;
b in REAL by XREAL_0:def_1;
then A2: -infty < b by XXREAL_0:12;
a < +infty by A1, XXREAL_0:9;
hence ].-infty,a.] /\ [.b,+infty.[ = [.b,a.] by A2, Th152; ::_thesis: verum
end;
theorem :: XXREAL_1:273
for s, p, r, q being ext-real number st s <= p holds
[.r,s.[ misses ].p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.[ misses ].p,q.[ )
assume A1: s <= p ; ::_thesis: [.r,s.[ misses ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.[ or not t in ].p,q.[ )
assume t in [.r,s.[ ; ::_thesis: not t in ].p,q.[
then t <= s by Th3;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:274
for s, p, r, q being ext-real number st s <= p holds
[.r,s.[ misses ].p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies [.r,s.[ misses ].p,q.] )
assume A1: s <= p ; ::_thesis: [.r,s.[ misses ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.[ or not t in ].p,q.] )
assume t in [.r,s.[ ; ::_thesis: not t in ].p,q.]
then t <= s by Th3;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:275
for s, p, r, q being ext-real number st s <= p holds
].r,s.[ misses ].p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.[ misses ].p,q.[ )
assume A1: s <= p ; ::_thesis: ].r,s.[ misses ].p,q.[
let t be real number ; :: according to MEMBERED:def_21 ::_thesis: ( not t in ].r,s.[ or not t in ].p,q.[ )
assume t in ].r,s.[ ; ::_thesis: not t in ].p,q.[
then t <= s by Th4;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; ::_thesis: verum
end;
theorem :: XXREAL_1:276
for s, p, r, q being ext-real number st s <= p holds
].r,s.[ misses ].p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s <= p implies ].r,s.[ misses ].p,q.] )
assume A1: s <= p ; ::_thesis: ].r,s.[ misses ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.[ or not t in ].p,q.] )
assume t in ].r,s.[ ; ::_thesis: not t in ].p,q.]
then t <= s by Th4;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.] by Th2; ::_thesis: verum
end;
theorem :: XXREAL_1:277
for s, p, r, q being ext-real number st s < p holds
[.r,s.] misses [.p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s < p implies [.r,s.] misses [.p,q.[ )
assume A1: s < p ; ::_thesis: [.r,s.] misses [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.] or not t in [.p,q.[ )
assume t in [.r,s.] ; ::_thesis: not t in [.p,q.[
then t <= s by Th1;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:278
for s, p, r, q being ext-real number st s < p holds
[.r,s.] misses [.p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s < p implies [.r,s.] misses [.p,q.] )
assume A1: s < p ; ::_thesis: [.r,s.] misses [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in [.r,s.] or not t in [.p,q.] )
assume t in [.r,s.] ; ::_thesis: not t in [.p,q.]
then t <= s by Th1;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:279
for s, p, r, q being ext-real number st s < p holds
].r,s.] misses [.p,q.[
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s < p implies ].r,s.] misses [.p,q.[ )
assume A1: s < p ; ::_thesis: ].r,s.] misses [.p,q.[
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.] or not t in [.p,q.[ )
assume t in ].r,s.] ; ::_thesis: not t in [.p,q.[
then t <= s by Th2;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; ::_thesis: verum
end;
theorem :: XXREAL_1:280
for s, p, r, q being ext-real number st s < p holds
].r,s.] misses [.p,q.]
proof
let s, p, r, q be ext-real number ; ::_thesis: ( s < p implies ].r,s.] misses [.p,q.] )
assume A1: s < p ; ::_thesis: ].r,s.] misses [.p,q.]
let t be ext-real number ; :: according to MEMBERED:def_20 ::_thesis: ( not t in ].r,s.] or not t in [.p,q.] )
assume t in ].r,s.] ; ::_thesis: not t in [.p,q.]
then t <= s by Th2;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.] by Th1; ::_thesis: verum
end;
theorem :: XXREAL_1:281
for t, s being ext-real number holds [.-infty,t.] \ [.-infty,s.] = ].s,t.] by Th182, XXREAL_0:5;
theorem :: XXREAL_1:282
for t, s being ext-real number holds [.-infty,t.[ \ [.-infty,s.] = ].s,t.[ by Th183, XXREAL_0:5;
theorem :: XXREAL_1:283
for t, s being ext-real number holds [.-infty,t.] \ [.-infty,s.] = ].s,t.] by Th186, XXREAL_0:5;
theorem :: XXREAL_1:284
for r, s being ext-real number holds [.r,+infty.] \ [.s,+infty.] = [.r,s.[ by Th190, XXREAL_0:3;
theorem :: XXREAL_1:285
for r, s being ext-real number holds ].r,+infty.] \ [.s,+infty.] = ].r,s.[ by Th191, XXREAL_0:3;
theorem :: XXREAL_1:286
for t being ext-real number
for s being real number holds [.-infty,t.] \ [.-infty,s.[ = [.s,t.]
proof
let t be ext-real number ; ::_thesis: for s being real number holds [.-infty,t.] \ [.-infty,s.[ = [.s,t.]
let s be real number ; ::_thesis: [.-infty,t.] \ [.-infty,s.[ = [.s,t.]
s in REAL by XREAL_0:def_1;
hence [.-infty,t.] \ [.-infty,s.[ = [.s,t.] by Th184, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:287
for t being ext-real number
for s being real number holds [.-infty,t.[ \ [.-infty,s.[ = [.s,t.[
proof
let t be ext-real number ; ::_thesis: for s being real number holds [.-infty,t.[ \ [.-infty,s.[ = [.s,t.[
let s be real number ; ::_thesis: [.-infty,t.[ \ [.-infty,s.[ = [.s,t.[
s in REAL by XREAL_0:def_1;
hence [.-infty,t.[ \ [.-infty,s.[ = [.s,t.[ by Th185, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:288
for t being ext-real number
for s being real number holds ].-infty,t.[ \ ].-infty,s.] = ].s,t.[
proof
let t be ext-real number ; ::_thesis: for s being real number holds ].-infty,t.[ \ ].-infty,s.] = ].s,t.[
let s be real number ; ::_thesis: ].-infty,t.[ \ ].-infty,s.] = ].s,t.[
s in REAL by XREAL_0:def_1;
hence ].-infty,t.[ \ ].-infty,s.] = ].s,t.[ by Th187, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:289
for t being ext-real number
for s being real number holds ].-infty,t.] \ ].-infty,s.[ = [.s,t.]
proof
let t be ext-real number ; ::_thesis: for s being real number holds ].-infty,t.] \ ].-infty,s.[ = [.s,t.]
let s be real number ; ::_thesis: ].-infty,t.] \ ].-infty,s.[ = [.s,t.]
s in REAL by XREAL_0:def_1;
hence ].-infty,t.] \ ].-infty,s.[ = [.s,t.] by Th188, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:290
for t being ext-real number
for s being real number holds ].-infty,t.[ \ ].-infty,s.[ = [.s,t.[
proof
let t be ext-real number ; ::_thesis: for s being real number holds ].-infty,t.[ \ ].-infty,s.[ = [.s,t.[
let s be real number ; ::_thesis: ].-infty,t.[ \ ].-infty,s.[ = [.s,t.[
s in REAL by XREAL_0:def_1;
hence ].-infty,t.[ \ ].-infty,s.[ = [.s,t.[ by Th189, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:291
for r being ext-real number
for s being real number holds [.r,+infty.] \ ].s,+infty.] = [.r,s.]
proof
let r be ext-real number ; ::_thesis: for s being real number holds [.r,+infty.] \ ].s,+infty.] = [.r,s.]
let s be real number ; ::_thesis: [.r,+infty.] \ ].s,+infty.] = [.r,s.]
s in REAL by XREAL_0:def_1;
hence [.r,+infty.] \ ].s,+infty.] = [.r,s.] by Th192, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:292
for r being ext-real number
for s being real number holds ].r,+infty.] \ ].s,+infty.] = ].r,s.]
proof
let r be ext-real number ; ::_thesis: for s being real number holds ].r,+infty.] \ ].s,+infty.] = ].r,s.]
let s be real number ; ::_thesis: ].r,+infty.] \ ].s,+infty.] = ].r,s.]
s in REAL by XREAL_0:def_1;
hence ].r,+infty.] \ ].s,+infty.] = ].r,s.] by Th193, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:293
for r being ext-real number
for s being real number holds [.r,+infty.[ \ [.s,+infty.[ = [.r,s.[
proof
let r be ext-real number ; ::_thesis: for s being real number holds [.r,+infty.[ \ [.s,+infty.[ = [.r,s.[
let s be real number ; ::_thesis: [.r,+infty.[ \ [.s,+infty.[ = [.r,s.[
s in REAL by XREAL_0:def_1;
hence [.r,+infty.[ \ [.s,+infty.[ = [.r,s.[ by Th194, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:294
for r being ext-real number
for s being real number holds ].r,+infty.[ \ [.s,+infty.[ = ].r,s.[
proof
let r be ext-real number ; ::_thesis: for s being real number holds ].r,+infty.[ \ [.s,+infty.[ = ].r,s.[
let s be real number ; ::_thesis: ].r,+infty.[ \ [.s,+infty.[ = ].r,s.[
s in REAL by XREAL_0:def_1;
hence ].r,+infty.[ \ [.s,+infty.[ = ].r,s.[ by Th195, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:295
for r being ext-real number
for s being real number holds [.r,+infty.[ \ ].s,+infty.[ = [.r,s.]
proof
let r be ext-real number ; ::_thesis: for s being real number holds [.r,+infty.[ \ ].s,+infty.[ = [.r,s.]
let s be real number ; ::_thesis: [.r,+infty.[ \ ].s,+infty.[ = [.r,s.]
s in REAL by XREAL_0:def_1;
hence [.r,+infty.[ \ ].s,+infty.[ = [.r,s.] by Th196, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:296
for r being ext-real number
for s being real number holds ].r,+infty.[ \ ].s,+infty.[ = ].r,s.]
proof
let r be ext-real number ; ::_thesis: for s being real number holds ].r,+infty.[ \ ].s,+infty.[ = ].r,s.]
let s be real number ; ::_thesis: ].r,+infty.[ \ ].s,+infty.[ = ].r,s.]
s in REAL by XREAL_0:def_1;
hence ].r,+infty.[ \ ].s,+infty.[ = ].r,s.] by Th197, XXREAL_0:9; ::_thesis: verum
end;
theorem Th297: :: XXREAL_1:297
for r, s, p, q being ext-real number st r < s & p < q holds
].r,q.[ \ ].p,s.[ = ].r,p.] \/ [.s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p < q implies ].r,q.[ \ ].p,s.[ = ].r,p.] \/ [.s,q.[ )
assume that
A1: r < s and
A2: p < q ; ::_thesis: ].r,q.[ \ ].p,s.[ = ].r,p.] \/ [.s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.[ \ ].p,s.[ or x in ].r,p.] \/ [.s,q.[ ) & ( not x in ].r,p.] \/ [.s,q.[ or x in ].r,q.[ \ ].p,s.[ ) )
thus ( x in ].r,q.[ \ ].p,s.[ implies x in ].r,p.] \/ [.s,q.[ ) ::_thesis: ( not x in ].r,p.] \/ [.s,q.[ or x in ].r,q.[ \ ].p,s.[ )
proof
assume A3: x in ].r,q.[ \ ].p,s.[ ; ::_thesis: x in ].r,p.] \/ [.s,q.[
then A4: not x in ].p,s.[ by XBOOLE_0:def_5;
A5: r < x by A3, Th4;
A6: x < q by A3, Th4;
( not p < x or not x < s ) by A4, Th4;
then ( x in ].r,p.] or x in [.s,q.[ ) by A5, A6, Th2, Th3;
hence x in ].r,p.] \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.] \/ [.s,q.[ ; ::_thesis: x in ].r,q.[ \ ].p,s.[
then ( x in ].r,p.] or x in [.s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x <= p ) or ( s <= x & x < q ) ) by Th2, Th3;
then A8: r < x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.[ by A8, Th4;
not x in ].p,s.[ by A7, Th4;
hence x in ].r,q.[ \ ].p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th298: :: XXREAL_1:298
for r, s, p, q being ext-real number st r <= s & p < q holds
[.r,q.[ \ ].p,s.[ = [.r,p.] \/ [.s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p < q implies [.r,q.[ \ ].p,s.[ = [.r,p.] \/ [.s,q.[ )
assume that
A1: r <= s and
A2: p < q ; ::_thesis: [.r,q.[ \ ].p,s.[ = [.r,p.] \/ [.s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.[ \ ].p,s.[ or x in [.r,p.] \/ [.s,q.[ ) & ( not x in [.r,p.] \/ [.s,q.[ or x in [.r,q.[ \ ].p,s.[ ) )
thus ( x in [.r,q.[ \ ].p,s.[ implies x in [.r,p.] \/ [.s,q.[ ) ::_thesis: ( not x in [.r,p.] \/ [.s,q.[ or x in [.r,q.[ \ ].p,s.[ )
proof
assume A3: x in [.r,q.[ \ ].p,s.[ ; ::_thesis: x in [.r,p.] \/ [.s,q.[
then A4: not x in ].p,s.[ by XBOOLE_0:def_5;
A5: r <= x by A3, Th3;
A6: x < q by A3, Th3;
( not p < x or not x < s ) by A4, Th4;
then ( x in [.r,p.] or x in [.s,q.[ ) by A5, A6, Th1, Th3;
hence x in [.r,p.] \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.] \/ [.s,q.[ ; ::_thesis: x in [.r,q.[ \ ].p,s.[
then ( x in [.r,p.] or x in [.s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x <= p ) or ( s <= x & x < q ) ) by Th1, Th3;
then A8: r <= x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.[ by A8, Th3;
not x in ].p,s.[ by A7, Th4;
hence x in [.r,q.[ \ ].p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th299: :: XXREAL_1:299
for r, s, p, q being ext-real number st r < s & p <= q holds
].r,q.] \ ].p,s.[ = ].r,p.] \/ [.s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p <= q implies ].r,q.] \ ].p,s.[ = ].r,p.] \/ [.s,q.] )
assume that
A1: r < s and
A2: p <= q ; ::_thesis: ].r,q.] \ ].p,s.[ = ].r,p.] \/ [.s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.] \ ].p,s.[ or x in ].r,p.] \/ [.s,q.] ) & ( not x in ].r,p.] \/ [.s,q.] or x in ].r,q.] \ ].p,s.[ ) )
thus ( x in ].r,q.] \ ].p,s.[ implies x in ].r,p.] \/ [.s,q.] ) ::_thesis: ( not x in ].r,p.] \/ [.s,q.] or x in ].r,q.] \ ].p,s.[ )
proof
assume A3: x in ].r,q.] \ ].p,s.[ ; ::_thesis: x in ].r,p.] \/ [.s,q.]
then A4: not x in ].p,s.[ by XBOOLE_0:def_5;
A5: r < x by A3, Th2;
A6: x <= q by A3, Th2;
( not p < x or not x < s ) by A4, Th4;
then ( x in ].r,p.] or x in [.s,q.] ) by A5, A6, Th1, Th2;
hence x in ].r,p.] \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.] \/ [.s,q.] ; ::_thesis: x in ].r,q.] \ ].p,s.[
then ( x in ].r,p.] or x in [.s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x <= p ) or ( s <= x & x <= q ) ) by Th1, Th2;
then A8: r < x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.] by A8, Th2;
not x in ].p,s.[ by A7, Th4;
hence x in ].r,q.] \ ].p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th300: :: XXREAL_1:300
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.] \ ].p,s.[ = [.r,p.] \/ [.s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.] \ ].p,s.[ = [.r,p.] \/ [.s,q.] )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.] \ ].p,s.[ = [.r,p.] \/ [.s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.] \ ].p,s.[ or x in [.r,p.] \/ [.s,q.] ) & ( not x in [.r,p.] \/ [.s,q.] or x in [.r,q.] \ ].p,s.[ ) )
thus ( x in [.r,q.] \ ].p,s.[ implies x in [.r,p.] \/ [.s,q.] ) ::_thesis: ( not x in [.r,p.] \/ [.s,q.] or x in [.r,q.] \ ].p,s.[ )
proof
assume A3: x in [.r,q.] \ ].p,s.[ ; ::_thesis: x in [.r,p.] \/ [.s,q.]
then A4: not x in ].p,s.[ by XBOOLE_0:def_5;
A5: r <= x by A3, Th1;
A6: x <= q by A3, Th1;
( not p < x or not x < s ) by A4, Th4;
then ( x in [.r,p.] or x in [.s,q.] ) by A5, A6, Th1;
hence x in [.r,p.] \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.] \/ [.s,q.] ; ::_thesis: x in [.r,q.] \ ].p,s.[
then ( x in [.r,p.] or x in [.s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x <= p ) or ( s <= x & x <= q ) ) by Th1;
then A8: r <= x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.] by A8, Th1;
not x in ].p,s.[ by A7, Th4;
hence x in [.r,q.] \ ].p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th301: :: XXREAL_1:301
for r, s, p, q being ext-real number st r < s & p <= q holds
].r,q.[ \ [.p,s.[ = ].r,p.[ \/ [.s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p <= q implies ].r,q.[ \ [.p,s.[ = ].r,p.[ \/ [.s,q.[ )
assume that
A1: r < s and
A2: p <= q ; ::_thesis: ].r,q.[ \ [.p,s.[ = ].r,p.[ \/ [.s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.[ \ [.p,s.[ or x in ].r,p.[ \/ [.s,q.[ ) & ( not x in ].r,p.[ \/ [.s,q.[ or x in ].r,q.[ \ [.p,s.[ ) )
thus ( x in ].r,q.[ \ [.p,s.[ implies x in ].r,p.[ \/ [.s,q.[ ) ::_thesis: ( not x in ].r,p.[ \/ [.s,q.[ or x in ].r,q.[ \ [.p,s.[ )
proof
assume A3: x in ].r,q.[ \ [.p,s.[ ; ::_thesis: x in ].r,p.[ \/ [.s,q.[
then A4: not x in [.p,s.[ by XBOOLE_0:def_5;
A5: r < x by A3, Th4;
A6: x < q by A3, Th4;
( not p <= x or not x < s ) by A4, Th3;
then ( x in ].r,p.[ or x in [.s,q.[ ) by A5, A6, Th3, Th4;
hence x in ].r,p.[ \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.[ \/ [.s,q.[ ; ::_thesis: x in ].r,q.[ \ [.p,s.[
then ( x in ].r,p.[ or x in [.s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x < p ) or ( s <= x & x < q ) ) by Th3, Th4;
then A8: r < x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.[ by A8, Th4;
not x in [.p,s.[ by A7, Th3;
hence x in ].r,q.[ \ [.p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th302: :: XXREAL_1:302
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.[ \ [.p,s.[ = [.r,p.[ \/ [.s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.[ \ [.p,s.[ = [.r,p.[ \/ [.s,q.[ )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.[ \ [.p,s.[ = [.r,p.[ \/ [.s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.[ \ [.p,s.[ or x in [.r,p.[ \/ [.s,q.[ ) & ( not x in [.r,p.[ \/ [.s,q.[ or x in [.r,q.[ \ [.p,s.[ ) )
thus ( x in [.r,q.[ \ [.p,s.[ implies x in [.r,p.[ \/ [.s,q.[ ) ::_thesis: ( not x in [.r,p.[ \/ [.s,q.[ or x in [.r,q.[ \ [.p,s.[ )
proof
assume A3: x in [.r,q.[ \ [.p,s.[ ; ::_thesis: x in [.r,p.[ \/ [.s,q.[
then A4: not x in [.p,s.[ by XBOOLE_0:def_5;
A5: r <= x by A3, Th3;
A6: x < q by A3, Th3;
( not p <= x or not x < s ) by A4, Th3;
then ( x in [.r,p.[ or x in [.s,q.[ ) by A5, A6, Th3;
hence x in [.r,p.[ \/ [.s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.[ \/ [.s,q.[ ; ::_thesis: x in [.r,q.[ \ [.p,s.[
then ( x in [.r,p.[ or x in [.s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x < p ) or ( s <= x & x < q ) ) by Th3;
then A8: r <= x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.[ by A8, Th3;
not x in [.p,s.[ by A7, Th3;
hence x in [.r,q.[ \ [.p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th303: :: XXREAL_1:303
for r, s, p, q being ext-real number st r < s & p <= q holds
].r,q.] \ [.p,s.[ = ].r,p.[ \/ [.s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p <= q implies ].r,q.] \ [.p,s.[ = ].r,p.[ \/ [.s,q.] )
assume that
A1: r < s and
A2: p <= q ; ::_thesis: ].r,q.] \ [.p,s.[ = ].r,p.[ \/ [.s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.] \ [.p,s.[ or x in ].r,p.[ \/ [.s,q.] ) & ( not x in ].r,p.[ \/ [.s,q.] or x in ].r,q.] \ [.p,s.[ ) )
thus ( x in ].r,q.] \ [.p,s.[ implies x in ].r,p.[ \/ [.s,q.] ) ::_thesis: ( not x in ].r,p.[ \/ [.s,q.] or x in ].r,q.] \ [.p,s.[ )
proof
assume A3: x in ].r,q.] \ [.p,s.[ ; ::_thesis: x in ].r,p.[ \/ [.s,q.]
then A4: not x in [.p,s.[ by XBOOLE_0:def_5;
A5: r < x by A3, Th2;
A6: x <= q by A3, Th2;
( not p <= x or not x < s ) by A4, Th3;
then ( x in ].r,p.[ or x in [.s,q.] ) by A5, A6, Th1, Th4;
hence x in ].r,p.[ \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.[ \/ [.s,q.] ; ::_thesis: x in ].r,q.] \ [.p,s.[
then ( x in ].r,p.[ or x in [.s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x < p ) or ( s <= x & x <= q ) ) by Th1, Th4;
then A8: r < x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.] by A8, Th2;
not x in [.p,s.[ by A7, Th3;
hence x in ].r,q.] \ [.p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th304: :: XXREAL_1:304
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.] \ [.p,s.[ = [.r,p.[ \/ [.s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.] \ [.p,s.[ = [.r,p.[ \/ [.s,q.] )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.] \ [.p,s.[ = [.r,p.[ \/ [.s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.] \ [.p,s.[ or x in [.r,p.[ \/ [.s,q.] ) & ( not x in [.r,p.[ \/ [.s,q.] or x in [.r,q.] \ [.p,s.[ ) )
thus ( x in [.r,q.] \ [.p,s.[ implies x in [.r,p.[ \/ [.s,q.] ) ::_thesis: ( not x in [.r,p.[ \/ [.s,q.] or x in [.r,q.] \ [.p,s.[ )
proof
assume A3: x in [.r,q.] \ [.p,s.[ ; ::_thesis: x in [.r,p.[ \/ [.s,q.]
then A4: not x in [.p,s.[ by XBOOLE_0:def_5;
A5: r <= x by A3, Th1;
A6: x <= q by A3, Th1;
( not p <= x or not x < s ) by A4, Th3;
then ( x in [.r,p.[ or x in [.s,q.] ) by A5, A6, Th1, Th3;
hence x in [.r,p.[ \/ [.s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.[ \/ [.s,q.] ; ::_thesis: x in [.r,q.] \ [.p,s.[
then ( x in [.r,p.[ or x in [.s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x < p ) or ( s <= x & x <= q ) ) by Th1, Th3;
then A8: r <= x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.] by A8, Th1;
not x in [.p,s.[ by A7, Th3;
hence x in [.r,q.] \ [.p,s.[ by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th305: :: XXREAL_1:305
for r, s, p, q being ext-real number st r < s & p < q holds
].r,q.[ \ ].p,s.] = ].r,p.] \/ ].s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p < q implies ].r,q.[ \ ].p,s.] = ].r,p.] \/ ].s,q.[ )
assume that
A1: r < s and
A2: p < q ; ::_thesis: ].r,q.[ \ ].p,s.] = ].r,p.] \/ ].s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.[ \ ].p,s.] or x in ].r,p.] \/ ].s,q.[ ) & ( not x in ].r,p.] \/ ].s,q.[ or x in ].r,q.[ \ ].p,s.] ) )
thus ( x in ].r,q.[ \ ].p,s.] implies x in ].r,p.] \/ ].s,q.[ ) ::_thesis: ( not x in ].r,p.] \/ ].s,q.[ or x in ].r,q.[ \ ].p,s.] )
proof
assume A3: x in ].r,q.[ \ ].p,s.] ; ::_thesis: x in ].r,p.] \/ ].s,q.[
then A4: not x in ].p,s.] by XBOOLE_0:def_5;
A5: r < x by A3, Th4;
A6: x < q by A3, Th4;
( not p < x or not x <= s ) by A4, Th2;
then ( x in ].r,p.] or x in ].s,q.[ ) by A5, A6, Th2, Th4;
hence x in ].r,p.] \/ ].s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.] \/ ].s,q.[ ; ::_thesis: x in ].r,q.[ \ ].p,s.]
then ( x in ].r,p.] or x in ].s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x <= p ) or ( s < x & x < q ) ) by Th2, Th4;
then A8: r < x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.[ by A8, Th4;
not x in ].p,s.] by A7, Th2;
hence x in ].r,q.[ \ ].p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th306: :: XXREAL_1:306
for r, s, p, q being ext-real number st r <= s & p < q holds
[.r,q.[ \ ].p,s.] = [.r,p.] \/ ].s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p < q implies [.r,q.[ \ ].p,s.] = [.r,p.] \/ ].s,q.[ )
assume that
A1: r <= s and
A2: p < q ; ::_thesis: [.r,q.[ \ ].p,s.] = [.r,p.] \/ ].s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.[ \ ].p,s.] or x in [.r,p.] \/ ].s,q.[ ) & ( not x in [.r,p.] \/ ].s,q.[ or x in [.r,q.[ \ ].p,s.] ) )
thus ( x in [.r,q.[ \ ].p,s.] implies x in [.r,p.] \/ ].s,q.[ ) ::_thesis: ( not x in [.r,p.] \/ ].s,q.[ or x in [.r,q.[ \ ].p,s.] )
proof
assume A3: x in [.r,q.[ \ ].p,s.] ; ::_thesis: x in [.r,p.] \/ ].s,q.[
then A4: not x in ].p,s.] by XBOOLE_0:def_5;
A5: r <= x by A3, Th3;
A6: x < q by A3, Th3;
( not p < x or not x <= s ) by A4, Th2;
then ( x in [.r,p.] or x in ].s,q.[ ) by A5, A6, Th1, Th4;
hence x in [.r,p.] \/ ].s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.] \/ ].s,q.[ ; ::_thesis: x in [.r,q.[ \ ].p,s.]
then ( x in [.r,p.] or x in ].s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x <= p ) or ( s < x & x < q ) ) by Th1, Th4;
then A8: r <= x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.[ by A8, Th3;
not x in ].p,s.] by A7, Th2;
hence x in [.r,q.[ \ ].p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th307: :: XXREAL_1:307
for r, s, p, q being ext-real number st r < s & p <= q holds
].r,q.] \ ].p,s.] = ].r,p.] \/ ].s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p <= q implies ].r,q.] \ ].p,s.] = ].r,p.] \/ ].s,q.] )
assume that
A1: r < s and
A2: p <= q ; ::_thesis: ].r,q.] \ ].p,s.] = ].r,p.] \/ ].s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.] \ ].p,s.] or x in ].r,p.] \/ ].s,q.] ) & ( not x in ].r,p.] \/ ].s,q.] or x in ].r,q.] \ ].p,s.] ) )
thus ( x in ].r,q.] \ ].p,s.] implies x in ].r,p.] \/ ].s,q.] ) ::_thesis: ( not x in ].r,p.] \/ ].s,q.] or x in ].r,q.] \ ].p,s.] )
proof
assume A3: x in ].r,q.] \ ].p,s.] ; ::_thesis: x in ].r,p.] \/ ].s,q.]
then A4: not x in ].p,s.] by XBOOLE_0:def_5;
A5: r < x by A3, Th2;
A6: x <= q by A3, Th2;
( not p < x or not x <= s ) by A4, Th2;
then ( x in ].r,p.] or x in ].s,q.] ) by A5, A6, Th2;
hence x in ].r,p.] \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.] \/ ].s,q.] ; ::_thesis: x in ].r,q.] \ ].p,s.]
then ( x in ].r,p.] or x in ].s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x <= p ) or ( s < x & x <= q ) ) by Th2;
then A8: r < x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.] by A8, Th2;
not x in ].p,s.] by A7, Th2;
hence x in ].r,q.] \ ].p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th308: :: XXREAL_1:308
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.] \ ].p,s.] = [.r,p.] \/ ].s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.] \ ].p,s.] = [.r,p.] \/ ].s,q.] )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.] \ ].p,s.] = [.r,p.] \/ ].s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.] \ ].p,s.] or x in [.r,p.] \/ ].s,q.] ) & ( not x in [.r,p.] \/ ].s,q.] or x in [.r,q.] \ ].p,s.] ) )
thus ( x in [.r,q.] \ ].p,s.] implies x in [.r,p.] \/ ].s,q.] ) ::_thesis: ( not x in [.r,p.] \/ ].s,q.] or x in [.r,q.] \ ].p,s.] )
proof
assume A3: x in [.r,q.] \ ].p,s.] ; ::_thesis: x in [.r,p.] \/ ].s,q.]
then A4: not x in ].p,s.] by XBOOLE_0:def_5;
A5: r <= x by A3, Th1;
A6: x <= q by A3, Th1;
( not p < x or not x <= s ) by A4, Th2;
then ( x in [.r,p.] or x in ].s,q.] ) by A5, A6, Th1, Th2;
hence x in [.r,p.] \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.] \/ ].s,q.] ; ::_thesis: x in [.r,q.] \ ].p,s.]
then ( x in [.r,p.] or x in ].s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x <= p ) or ( s < x & x <= q ) ) by Th1, Th2;
then A8: r <= x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.] by A8, Th1;
not x in ].p,s.] by A7, Th2;
hence x in [.r,q.] \ ].p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th309: :: XXREAL_1:309
for r, s, p, q being ext-real number st r <= s & p <= q holds
].r,q.[ \ [.p,s.] = ].r,p.[ \/ ].s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies ].r,q.[ \ [.p,s.] = ].r,p.[ \/ ].s,q.[ )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: ].r,q.[ \ [.p,s.] = ].r,p.[ \/ ].s,q.[
let x be real number ; :: according to MEMBERED:def_15 ::_thesis: ( ( not x in ].r,q.[ \ [.p,s.] or x in ].r,p.[ \/ ].s,q.[ ) & ( not x in ].r,p.[ \/ ].s,q.[ or x in ].r,q.[ \ [.p,s.] ) )
thus ( x in ].r,q.[ \ [.p,s.] implies x in ].r,p.[ \/ ].s,q.[ ) ::_thesis: ( not x in ].r,p.[ \/ ].s,q.[ or x in ].r,q.[ \ [.p,s.] )
proof
assume A3: x in ].r,q.[ \ [.p,s.] ; ::_thesis: x in ].r,p.[ \/ ].s,q.[
then A4: not x in [.p,s.] by XBOOLE_0:def_5;
A5: r < x by A3, Th4;
A6: x < q by A3, Th4;
( not p <= x or not x <= s ) by A4, Th1;
then ( x in ].r,p.[ or x in ].s,q.[ ) by A5, A6, Th4;
hence x in ].r,p.[ \/ ].s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.[ \/ ].s,q.[ ; ::_thesis: x in ].r,q.[ \ [.p,s.]
then ( x in ].r,p.[ or x in ].s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x < p ) or ( s < x & x < q ) ) by Th4;
then A8: r < x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.[ by A8, Th4;
not x in [.p,s.] by A7, Th1;
hence x in ].r,q.[ \ [.p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th310: :: XXREAL_1:310
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.[ \ [.p,s.] = [.r,p.[ \/ ].s,q.[
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.[ \ [.p,s.] = [.r,p.[ \/ ].s,q.[ )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.[ \ [.p,s.] = [.r,p.[ \/ ].s,q.[
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.[ \ [.p,s.] or x in [.r,p.[ \/ ].s,q.[ ) & ( not x in [.r,p.[ \/ ].s,q.[ or x in [.r,q.[ \ [.p,s.] ) )
thus ( x in [.r,q.[ \ [.p,s.] implies x in [.r,p.[ \/ ].s,q.[ ) ::_thesis: ( not x in [.r,p.[ \/ ].s,q.[ or x in [.r,q.[ \ [.p,s.] )
proof
assume A3: x in [.r,q.[ \ [.p,s.] ; ::_thesis: x in [.r,p.[ \/ ].s,q.[
then A4: not x in [.p,s.] by XBOOLE_0:def_5;
A5: r <= x by A3, Th3;
A6: x < q by A3, Th3;
( not p <= x or not x <= s ) by A4, Th1;
then ( x in [.r,p.[ or x in ].s,q.[ ) by A5, A6, Th3, Th4;
hence x in [.r,p.[ \/ ].s,q.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.[ \/ ].s,q.[ ; ::_thesis: x in [.r,q.[ \ [.p,s.]
then ( x in [.r,p.[ or x in ].s,q.[ ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x < p ) or ( s < x & x < q ) ) by Th3, Th4;
then A8: r <= x by A1, XXREAL_0:2;
x < q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.[ by A8, Th3;
not x in [.p,s.] by A7, Th1;
hence x in [.r,q.[ \ [.p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th311: :: XXREAL_1:311
for r, s, p, q being ext-real number st r < s & p <= q holds
].r,q.] \ [.p,s.] = ].r,p.[ \/ ].s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r < s & p <= q implies ].r,q.] \ [.p,s.] = ].r,p.[ \/ ].s,q.] )
assume that
A1: r < s and
A2: p <= q ; ::_thesis: ].r,q.] \ [.p,s.] = ].r,p.[ \/ ].s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in ].r,q.] \ [.p,s.] or x in ].r,p.[ \/ ].s,q.] ) & ( not x in ].r,p.[ \/ ].s,q.] or x in ].r,q.] \ [.p,s.] ) )
thus ( x in ].r,q.] \ [.p,s.] implies x in ].r,p.[ \/ ].s,q.] ) ::_thesis: ( not x in ].r,p.[ \/ ].s,q.] or x in ].r,q.] \ [.p,s.] )
proof
assume A3: x in ].r,q.] \ [.p,s.] ; ::_thesis: x in ].r,p.[ \/ ].s,q.]
then A4: not x in [.p,s.] by XBOOLE_0:def_5;
A5: r < x by A3, Th2;
A6: x <= q by A3, Th2;
( not p <= x or not x <= s ) by A4, Th1;
then ( x in ].r,p.[ or x in ].s,q.] ) by A5, A6, Th2, Th4;
hence x in ].r,p.[ \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in ].r,p.[ \/ ].s,q.] ; ::_thesis: x in ].r,q.] \ [.p,s.]
then ( x in ].r,p.[ or x in ].s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r < x & x < p ) or ( s < x & x <= q ) ) by Th2, Th4;
then A8: r < x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.] by A8, Th2;
not x in [.p,s.] by A7, Th1;
hence x in ].r,q.] \ [.p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th312: :: XXREAL_1:312
for r, s, p, q being ext-real number st r <= s & p <= q holds
[.r,q.] \ [.p,s.] = [.r,p.[ \/ ].s,q.]
proof
let r, s, p, q be ext-real number ; ::_thesis: ( r <= s & p <= q implies [.r,q.] \ [.p,s.] = [.r,p.[ \/ ].s,q.] )
assume that
A1: r <= s and
A2: p <= q ; ::_thesis: [.r,q.] \ [.p,s.] = [.r,p.[ \/ ].s,q.]
let x be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not x in [.r,q.] \ [.p,s.] or x in [.r,p.[ \/ ].s,q.] ) & ( not x in [.r,p.[ \/ ].s,q.] or x in [.r,q.] \ [.p,s.] ) )
thus ( x in [.r,q.] \ [.p,s.] implies x in [.r,p.[ \/ ].s,q.] ) ::_thesis: ( not x in [.r,p.[ \/ ].s,q.] or x in [.r,q.] \ [.p,s.] )
proof
assume A3: x in [.r,q.] \ [.p,s.] ; ::_thesis: x in [.r,p.[ \/ ].s,q.]
then A4: not x in [.p,s.] by XBOOLE_0:def_5;
A5: r <= x by A3, Th1;
A6: x <= q by A3, Th1;
( not p <= x or not x <= s ) by A4, Th1;
then ( x in [.r,p.[ or x in ].s,q.] ) by A5, A6, Th2, Th3;
hence x in [.r,p.[ \/ ].s,q.] by XBOOLE_0:def_3; ::_thesis: verum
end;
assume x in [.r,p.[ \/ ].s,q.] ; ::_thesis: x in [.r,q.] \ [.p,s.]
then ( x in [.r,p.[ or x in ].s,q.] ) by XBOOLE_0:def_3;
then A7: ( ( r <= x & x < p ) or ( s < x & x <= q ) ) by Th2, Th3;
then A8: r <= x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in [.r,q.] by A8, Th1;
not x in [.p,s.] by A7, Th1;
hence x in [.r,q.] \ [.p,s.] by A9, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th313: :: XXREAL_1:313
for r, p, q being ext-real number st r <= p & p <= q holds
].r,q.[ \ {p} = ].r,p.[ \/ ].p,q.[
proof
let r, p, q be ext-real number ; ::_thesis: ( r <= p & p <= q implies ].r,q.[ \ {p} = ].r,p.[ \/ ].p,q.[ )
[.p,p.] = {p} by Th17;
hence ( r <= p & p <= q implies ].r,q.[ \ {p} = ].r,p.[ \/ ].p,q.[ ) by Th309; ::_thesis: verum
end;
theorem Th314: :: XXREAL_1:314
for r, p, q being ext-real number st r <= p & p <= q holds
[.r,q.[ \ {p} = [.r,p.[ \/ ].p,q.[
proof
let r, p, q be ext-real number ; ::_thesis: ( r <= p & p <= q implies [.r,q.[ \ {p} = [.r,p.[ \/ ].p,q.[ )
[.p,p.] = {p} by Th17;
hence ( r <= p & p <= q implies [.r,q.[ \ {p} = [.r,p.[ \/ ].p,q.[ ) by Th310; ::_thesis: verum
end;
theorem Th315: :: XXREAL_1:315
for r, p, q being ext-real number st r < p & p <= q holds
].r,q.] \ {p} = ].r,p.[ \/ ].p,q.]
proof
let r, p, q be ext-real number ; ::_thesis: ( r < p & p <= q implies ].r,q.] \ {p} = ].r,p.[ \/ ].p,q.] )
[.p,p.] = {p} by Th17;
hence ( r < p & p <= q implies ].r,q.] \ {p} = ].r,p.[ \/ ].p,q.] ) by Th311; ::_thesis: verum
end;
theorem Th316: :: XXREAL_1:316
for r, p, q being ext-real number st r <= p & p <= q holds
[.r,q.] \ {p} = [.r,p.[ \/ ].p,q.]
proof
let r, p, q be ext-real number ; ::_thesis: ( r <= p & p <= q implies [.r,q.] \ {p} = [.r,p.[ \/ ].p,q.] )
[.p,p.] = {p} by Th17;
hence ( r <= p & p <= q implies [.r,q.] \ {p} = [.r,p.[ \/ ].p,q.] ) by Th312; ::_thesis: verum
end;
theorem Th317: :: XXREAL_1:317
for r, q, p being ext-real number st r < q & p <= q holds
].r,q.] \ ].p,q.[ = ].r,p.] \/ {q}
proof
let r, q, p be ext-real number ; ::_thesis: ( r < q & p <= q implies ].r,q.] \ ].p,q.[ = ].r,p.] \/ {q} )
[.q,q.] = {q} by Th17;
hence ( r < q & p <= q implies ].r,q.] \ ].p,q.[ = ].r,p.] \/ {q} ) by Th299; ::_thesis: verum
end;
theorem Th318: :: XXREAL_1:318
for r, q, p being ext-real number st r <= q & p <= q holds
[.r,q.] \ ].p,q.[ = [.r,p.] \/ {q}
proof
let r, q, p be ext-real number ; ::_thesis: ( r <= q & p <= q implies [.r,q.] \ ].p,q.[ = [.r,p.] \/ {q} )
[.q,q.] = {q} by Th17;
hence ( r <= q & p <= q implies [.r,q.] \ ].p,q.[ = [.r,p.] \/ {q} ) by Th300; ::_thesis: verum
end;
theorem Th319: :: XXREAL_1:319
for r, q, p being ext-real number st r < q & p <= q holds
].r,q.] \ [.p,q.[ = ].r,p.[ \/ {q}
proof
let r, q, p be ext-real number ; ::_thesis: ( r < q & p <= q implies ].r,q.] \ [.p,q.[ = ].r,p.[ \/ {q} )
[.q,q.] = {q} by Th17;
hence ( r < q & p <= q implies ].r,q.] \ [.p,q.[ = ].r,p.[ \/ {q} ) by Th303; ::_thesis: verum
end;
theorem Th320: :: XXREAL_1:320
for r, q, p being ext-real number st r <= q & p <= q holds
[.r,q.] \ [.p,q.[ = [.r,p.[ \/ {q}
proof
let r, q, p be ext-real number ; ::_thesis: ( r <= q & p <= q implies [.r,q.] \ [.p,q.[ = [.r,p.[ \/ {q} )
[.q,q.] = {q} by Th17;
hence ( r <= q & p <= q implies [.r,q.] \ [.p,q.[ = [.r,p.[ \/ {q} ) by Th304; ::_thesis: verum
end;
theorem Th321: :: XXREAL_1:321
for p, s, q being ext-real number st p <= s & p < q holds
[.p,q.[ \ ].p,s.[ = {p} \/ [.s,q.[
proof
let p, s, q be ext-real number ; ::_thesis: ( p <= s & p < q implies [.p,q.[ \ ].p,s.[ = {p} \/ [.s,q.[ )
[.p,p.] = {p} by Th17;
hence ( p <= s & p < q implies [.p,q.[ \ ].p,s.[ = {p} \/ [.s,q.[ ) by Th298; ::_thesis: verum
end;
theorem Th322: :: XXREAL_1:322
for p, s, q being ext-real number st p <= s & p <= q holds
[.p,q.] \ ].p,s.[ = {p} \/ [.s,q.]
proof
let p, s, q be ext-real number ; ::_thesis: ( p <= s & p <= q implies [.p,q.] \ ].p,s.[ = {p} \/ [.s,q.] )
[.p,p.] = {p} by Th17;
hence ( p <= s & p <= q implies [.p,q.] \ ].p,s.[ = {p} \/ [.s,q.] ) by Th300; ::_thesis: verum
end;
theorem Th323: :: XXREAL_1:323
for p, s, q being ext-real number st p <= s & p < q holds
[.p,q.[ \ ].p,s.] = {p} \/ ].s,q.[
proof
let p, s, q be ext-real number ; ::_thesis: ( p <= s & p < q implies [.p,q.[ \ ].p,s.] = {p} \/ ].s,q.[ )
[.p,p.] = {p} by Th17;
hence ( p <= s & p < q implies [.p,q.[ \ ].p,s.] = {p} \/ ].s,q.[ ) by Th306; ::_thesis: verum
end;
theorem Th324: :: XXREAL_1:324
for p, s, q being ext-real number st p <= s & p <= q holds
[.p,q.] \ ].p,s.] = {p} \/ ].s,q.]
proof
let p, s, q be ext-real number ; ::_thesis: ( p <= s & p <= q implies [.p,q.] \ ].p,s.] = {p} \/ ].s,q.] )
[.p,p.] = {p} by Th17;
hence ( p <= s & p <= q implies [.p,q.] \ ].p,s.] = {p} \/ ].s,q.] ) by Th308; ::_thesis: verum
end;
theorem Th325: :: XXREAL_1:325
for p, q, s being ext-real number st p < q holds
[.-infty,q.[ \ ].p,s.[ = [.-infty,p.] \/ [.s,q.[ by Th298, XXREAL_0:5;
theorem Th326: :: XXREAL_1:326
for p, q, s being ext-real number st p <= q holds
[.-infty,q.] \ ].p,s.[ = [.-infty,p.] \/ [.s,q.]
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ ].p,s.[ = [.-infty,p.] \/ [.s,q.] )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ ].p,s.[ = [.-infty,p.] \/ [.s,q.] ) by Th300; ::_thesis: verum
end;
theorem Th327: :: XXREAL_1:327
for p, q, s being ext-real number st p <= q holds
[.-infty,q.[ \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.[
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.[ \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.[ )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.[ \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.[ ) by Th302; ::_thesis: verum
end;
theorem Th328: :: XXREAL_1:328
for p, q, s being ext-real number st p <= q holds
[.-infty,q.] \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.]
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.] )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ [.p,s.[ = [.-infty,p.[ \/ [.s,q.] ) by Th304; ::_thesis: verum
end;
theorem Th329: :: XXREAL_1:329
for p, q, s being ext-real number st p < q holds
[.-infty,q.[ \ ].p,s.] = [.-infty,p.] \/ ].s,q.[ by Th306, XXREAL_0:5;
theorem Th330: :: XXREAL_1:330
for p, q, s being ext-real number st p <= q holds
[.-infty,q.] \ ].p,s.] = [.-infty,p.] \/ ].s,q.]
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ ].p,s.] = [.-infty,p.] \/ ].s,q.] )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ ].p,s.] = [.-infty,p.] \/ ].s,q.] ) by Th308; ::_thesis: verum
end;
theorem Th331: :: XXREAL_1:331
for p, q, s being ext-real number st p <= q holds
[.-infty,q.[ \ [.p,s.] = [.-infty,p.[ \/ ].s,q.[
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.[ \ [.p,s.] = [.-infty,p.[ \/ ].s,q.[ )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.[ \ [.p,s.] = [.-infty,p.[ \/ ].s,q.[ ) by Th310; ::_thesis: verum
end;
theorem Th332: :: XXREAL_1:332
for p, q, s being ext-real number st p <= q holds
[.-infty,q.] \ [.p,s.] = [.-infty,p.[ \/ ].s,q.]
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ [.p,s.] = [.-infty,p.[ \/ ].s,q.] )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ [.p,s.] = [.-infty,p.[ \/ ].s,q.] ) by Th312; ::_thesis: verum
end;
theorem Th333: :: XXREAL_1:333
for p, q being ext-real number
for s being real number st p < q holds
].-infty,q.[ \ ].p,s.[ = ].-infty,p.] \/ [.s,q.[
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p < q holds
].-infty,q.[ \ ].p,s.[ = ].-infty,p.] \/ [.s,q.[
let s be real number ; ::_thesis: ( p < q implies ].-infty,q.[ \ ].p,s.[ = ].-infty,p.] \/ [.s,q.[ )
s in REAL by XREAL_0:def_1;
then -infty < s by XXREAL_0:12;
hence ( p < q implies ].-infty,q.[ \ ].p,s.[ = ].-infty,p.] \/ [.s,q.[ ) by Th297; ::_thesis: verum
end;
theorem Th334: :: XXREAL_1:334
for p, q being ext-real number
for s being real number st p <= q holds
].-infty,q.] \ ].p,s.[ = ].-infty,p.] \/ [.s,q.]
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p <= q holds
].-infty,q.] \ ].p,s.[ = ].-infty,p.] \/ [.s,q.]
let s be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ ].p,s.[ = ].-infty,p.] \/ [.s,q.] )
s in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ ].p,s.[ = ].-infty,p.] \/ [.s,q.] ) by Th299, XXREAL_0:12; ::_thesis: verum
end;
theorem Th335: :: XXREAL_1:335
for p, q being ext-real number
for s being real number st p <= q holds
].-infty,q.[ \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.[
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p <= q holds
].-infty,q.[ \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.[
let s be real number ; ::_thesis: ( p <= q implies ].-infty,q.[ \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.[ )
s in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.[ \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.[ ) by Th301, XXREAL_0:12; ::_thesis: verum
end;
theorem Th336: :: XXREAL_1:336
for p, q being ext-real number
for s being real number st p <= q holds
].-infty,q.] \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.]
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p <= q holds
].-infty,q.] \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.]
let s be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.] )
s in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ [.p,s.[ = ].-infty,p.[ \/ [.s,q.] ) by Th303, XXREAL_0:12; ::_thesis: verum
end;
theorem Th337: :: XXREAL_1:337
for p, q being ext-real number
for s being real number st p < q holds
].-infty,q.[ \ ].p,s.] = ].-infty,p.] \/ ].s,q.[
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p < q holds
].-infty,q.[ \ ].p,s.] = ].-infty,p.] \/ ].s,q.[
let s be real number ; ::_thesis: ( p < q implies ].-infty,q.[ \ ].p,s.] = ].-infty,p.] \/ ].s,q.[ )
s in REAL by XREAL_0:def_1;
then -infty < s by XXREAL_0:12;
hence ( p < q implies ].-infty,q.[ \ ].p,s.] = ].-infty,p.] \/ ].s,q.[ ) by Th305; ::_thesis: verum
end;
theorem Th338: :: XXREAL_1:338
for p, q being ext-real number
for s being real number st p <= q holds
].-infty,q.] \ ].p,s.] = ].-infty,p.] \/ ].s,q.]
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p <= q holds
].-infty,q.] \ ].p,s.] = ].-infty,p.] \/ ].s,q.]
let s be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ ].p,s.] = ].-infty,p.] \/ ].s,q.] )
s in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ ].p,s.] = ].-infty,p.] \/ ].s,q.] ) by Th307, XXREAL_0:12; ::_thesis: verum
end;
theorem Th339: :: XXREAL_1:339
for p, q, s being ext-real number st p <= q holds
].-infty,q.[ \ [.p,s.] = ].-infty,p.[ \/ ].s,q.[
proof
let p, q, s be ext-real number ; ::_thesis: ( p <= q implies ].-infty,q.[ \ [.p,s.] = ].-infty,p.[ \/ ].s,q.[ )
-infty <= s by XXREAL_0:5;
hence ( p <= q implies ].-infty,q.[ \ [.p,s.] = ].-infty,p.[ \/ ].s,q.[ ) by Th309; ::_thesis: verum
end;
theorem Th340: :: XXREAL_1:340
for p, q being ext-real number
for s being real number st p <= q holds
].-infty,q.] \ [.p,s.] = ].-infty,p.[ \/ ].s,q.]
proof
let p, q be ext-real number ; ::_thesis: for s being real number st p <= q holds
].-infty,q.] \ [.p,s.] = ].-infty,p.[ \/ ].s,q.]
let s be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ [.p,s.] = ].-infty,p.[ \/ ].s,q.] )
s in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ [.p,s.] = ].-infty,p.[ \/ ].s,q.] ) by Th311, XXREAL_0:12; ::_thesis: verum
end;
theorem Th341: :: XXREAL_1:341
for p, q being ext-real number st p <= q holds
[.-infty,q.[ \ {p} = [.-infty,p.[ \/ ].p,q.[
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.[ \ {p} = [.-infty,p.[ \/ ].p,q.[ )
-infty <= p by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.[ \ {p} = [.-infty,p.[ \/ ].p,q.[ ) by Th314; ::_thesis: verum
end;
theorem Th342: :: XXREAL_1:342
for p, q being ext-real number st p <= q holds
[.-infty,q.] \ {p} = [.-infty,p.[ \/ ].p,q.]
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ {p} = [.-infty,p.[ \/ ].p,q.] )
-infty <= p by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ {p} = [.-infty,p.[ \/ ].p,q.] ) by Th316; ::_thesis: verum
end;
theorem :: XXREAL_1:343
for p, q being ext-real number st p <= q holds
[.-infty,q.] \ ].p,q.[ = [.-infty,p.] \/ {q}
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ ].p,q.[ = [.-infty,p.] \/ {q} )
-infty <= q by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ ].p,q.[ = [.-infty,p.] \/ {q} ) by Th318; ::_thesis: verum
end;
theorem :: XXREAL_1:344
for p, q being ext-real number st p <= q holds
[.-infty,q.] \ [.p,q.[ = [.-infty,p.[ \/ {q}
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies [.-infty,q.] \ [.p,q.[ = [.-infty,p.[ \/ {q} )
-infty <= q by XXREAL_0:5;
hence ( p <= q implies [.-infty,q.] \ [.p,q.[ = [.-infty,p.[ \/ {q} ) by Th320; ::_thesis: verum
end;
theorem :: XXREAL_1:345
for q, s being ext-real number holds [.-infty,q.] \ ].-infty,s.[ = {-infty} \/ [.s,q.]
proof
let q, s be ext-real number ; ::_thesis: [.-infty,q.] \ ].-infty,s.[ = {-infty} \/ [.s,q.]
A1: -infty <= s by XXREAL_0:5;
-infty <= q by XXREAL_0:5;
hence [.-infty,q.] \ ].-infty,s.[ = {-infty} \/ [.s,q.] by A1, Th322; ::_thesis: verum
end;
theorem :: XXREAL_1:346
for q, s being ext-real number holds [.-infty,q.] \ ].-infty,s.] = {-infty} \/ ].s,q.]
proof
let q, s be ext-real number ; ::_thesis: [.-infty,q.] \ ].-infty,s.] = {-infty} \/ ].s,q.]
A1: -infty <= s by XXREAL_0:5;
-infty <= q by XXREAL_0:5;
hence [.-infty,q.] \ ].-infty,s.] = {-infty} \/ ].s,q.] by A1, Th324; ::_thesis: verum
end;
theorem :: XXREAL_1:347
for s being ext-real number
for q being real number holds [.-infty,q.[ \ ].-infty,s.[ = {-infty} \/ [.s,q.[
proof
let s be ext-real number ; ::_thesis: for q being real number holds [.-infty,q.[ \ ].-infty,s.[ = {-infty} \/ [.s,q.[
let q be real number ; ::_thesis: [.-infty,q.[ \ ].-infty,s.[ = {-infty} \/ [.s,q.[
A1: q in REAL by XREAL_0:def_1;
-infty <= s by XXREAL_0:5;
hence [.-infty,q.[ \ ].-infty,s.[ = {-infty} \/ [.s,q.[ by A1, Th321, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:348
for s being ext-real number
for q being real number holds [.-infty,q.[ \ ].-infty,s.] = {-infty} \/ ].s,q.[
proof
let s be ext-real number ; ::_thesis: for q being real number holds [.-infty,q.[ \ ].-infty,s.] = {-infty} \/ ].s,q.[
let q be real number ; ::_thesis: [.-infty,q.[ \ ].-infty,s.] = {-infty} \/ ].s,q.[
A1: q in REAL by XREAL_0:def_1;
-infty <= s by XXREAL_0:5;
hence [.-infty,q.[ \ ].-infty,s.] = {-infty} \/ ].s,q.[ by A1, Th323, XXREAL_0:12; ::_thesis: verum
end;
theorem Th349: :: XXREAL_1:349
for p, q being ext-real number st p <= q holds
].-infty,q.[ \ {p} = ].-infty,p.[ \/ ].p,q.[
proof
let p, q be ext-real number ; ::_thesis: ( p <= q implies ].-infty,q.[ \ {p} = ].-infty,p.[ \/ ].p,q.[ )
-infty <= p by XXREAL_0:5;
hence ( p <= q implies ].-infty,q.[ \ {p} = ].-infty,p.[ \/ ].p,q.[ ) by Th313; ::_thesis: verum
end;
theorem Th350: :: XXREAL_1:350
for q being ext-real number
for p being real number st p <= q holds
].-infty,q.] \ {p} = ].-infty,p.[ \/ ].p,q.]
proof
let q be ext-real number ; ::_thesis: for p being real number st p <= q holds
].-infty,q.] \ {p} = ].-infty,p.[ \/ ].p,q.]
let p be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ {p} = ].-infty,p.[ \/ ].p,q.] )
p in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ {p} = ].-infty,p.[ \/ ].p,q.] ) by Th315, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:351
for p being ext-real number
for q being real number st p <= q holds
].-infty,q.] \ ].p,q.[ = ].-infty,p.] \/ {q}
proof
let p be ext-real number ; ::_thesis: for q being real number st p <= q holds
].-infty,q.] \ ].p,q.[ = ].-infty,p.] \/ {q}
let q be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ ].p,q.[ = ].-infty,p.] \/ {q} )
q in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ ].p,q.[ = ].-infty,p.] \/ {q} ) by Th317, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:352
for p being ext-real number
for q being real number st p <= q holds
].-infty,q.] \ [.p,q.[ = ].-infty,p.[ \/ {q}
proof
let p be ext-real number ; ::_thesis: for q being real number st p <= q holds
].-infty,q.] \ [.p,q.[ = ].-infty,p.[ \/ {q}
let q be real number ; ::_thesis: ( p <= q implies ].-infty,q.] \ [.p,q.[ = ].-infty,p.[ \/ {q} )
q in REAL by XREAL_0:def_1;
hence ( p <= q implies ].-infty,q.] \ [.p,q.[ = ].-infty,p.[ \/ {q} ) by Th319, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:353
for r, s, p being ext-real number st r < s holds
].r,+infty.] \ ].p,s.[ = ].r,p.] \/ [.s,+infty.] by Th299, XXREAL_0:3;
theorem :: XXREAL_1:354
for r, s, p being ext-real number st r <= s holds
[.r,+infty.] \ ].p,s.[ = [.r,p.] \/ [.s,+infty.]
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.] \ ].p,s.[ = [.r,p.] \/ [.s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.] \ ].p,s.[ = [.r,p.] \/ [.s,+infty.] ) by Th300; ::_thesis: verum
end;
theorem :: XXREAL_1:355
for r, s, p being ext-real number st r < s holds
].r,+infty.[ \ [.p,s.[ = ].r,p.[ \/ [.s,+infty.[ by Th301, XXREAL_0:3;
theorem :: XXREAL_1:356
for r, s, p being ext-real number st r <= s holds
[.r,+infty.[ \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.[
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.[ \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.[ )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.[ \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.[ ) by Th302; ::_thesis: verum
end;
theorem :: XXREAL_1:357
for r, s, p being ext-real number st r < s holds
].r,+infty.] \ [.p,s.[ = ].r,p.[ \/ [.s,+infty.] by Th303, XXREAL_0:3;
theorem :: XXREAL_1:358
for r, s, p being ext-real number st r <= s holds
[.r,+infty.] \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.]
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.] \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.] \ [.p,s.[ = [.r,p.[ \/ [.s,+infty.] ) by Th304; ::_thesis: verum
end;
theorem :: XXREAL_1:359
for r, s, p being ext-real number st r < s holds
].r,+infty.] \ ].p,s.] = ].r,p.] \/ ].s,+infty.] by Th307, XXREAL_0:3;
theorem :: XXREAL_1:360
for r, s, p being ext-real number st r <= s holds
[.r,+infty.] \ ].p,s.] = [.r,p.] \/ ].s,+infty.]
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.] \ ].p,s.] = [.r,p.] \/ ].s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.] \ ].p,s.] = [.r,p.] \/ ].s,+infty.] ) by Th308; ::_thesis: verum
end;
theorem :: XXREAL_1:361
for r, s, p being ext-real number st r <= s holds
].r,+infty.[ \ [.p,s.] = ].r,p.[ \/ ].s,+infty.[
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies ].r,+infty.[ \ [.p,s.] = ].r,p.[ \/ ].s,+infty.[ )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies ].r,+infty.[ \ [.p,s.] = ].r,p.[ \/ ].s,+infty.[ ) by Th309; ::_thesis: verum
end;
theorem :: XXREAL_1:362
for r, s, p being ext-real number st r <= s holds
[.r,+infty.[ \ [.p,s.] = [.r,p.[ \/ ].s,+infty.[
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.[ \ [.p,s.] = [.r,p.[ \/ ].s,+infty.[ )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.[ \ [.p,s.] = [.r,p.[ \/ ].s,+infty.[ ) by Th310; ::_thesis: verum
end;
theorem :: XXREAL_1:363
for r, s, p being ext-real number st r < s holds
].r,+infty.] \ [.p,s.] = ].r,p.[ \/ ].s,+infty.] by Th311, XXREAL_0:3;
theorem :: XXREAL_1:364
for r, s, p being ext-real number st r <= s holds
[.r,+infty.] \ [.p,s.] = [.r,p.[ \/ ].s,+infty.]
proof
let r, s, p be ext-real number ; ::_thesis: ( r <= s implies [.r,+infty.] \ [.p,s.] = [.r,p.[ \/ ].s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( r <= s implies [.r,+infty.] \ [.p,s.] = [.r,p.[ \/ ].s,+infty.] ) by Th312; ::_thesis: verum
end;
theorem :: XXREAL_1:365
for r, p being ext-real number st r <= p holds
].r,+infty.[ \ {p} = ].r,p.[ \/ ].p,+infty.[
proof
let r, p be ext-real number ; ::_thesis: ( r <= p implies ].r,+infty.[ \ {p} = ].r,p.[ \/ ].p,+infty.[ )
p <= +infty by XXREAL_0:3;
hence ( r <= p implies ].r,+infty.[ \ {p} = ].r,p.[ \/ ].p,+infty.[ ) by Th313; ::_thesis: verum
end;
theorem :: XXREAL_1:366
for r, p being ext-real number st r <= p holds
[.r,+infty.[ \ {p} = [.r,p.[ \/ ].p,+infty.[
proof
let r, p be ext-real number ; ::_thesis: ( r <= p implies [.r,+infty.[ \ {p} = [.r,p.[ \/ ].p,+infty.[ )
p <= +infty by XXREAL_0:3;
hence ( r <= p implies [.r,+infty.[ \ {p} = [.r,p.[ \/ ].p,+infty.[ ) by Th314; ::_thesis: verum
end;
theorem :: XXREAL_1:367
for r, p being ext-real number st r < p holds
].r,+infty.] \ {p} = ].r,p.[ \/ ].p,+infty.] by Th315, XXREAL_0:3;
theorem :: XXREAL_1:368
for r, p being ext-real number st r <= p holds
[.r,+infty.] \ {p} = [.r,p.[ \/ ].p,+infty.]
proof
let r, p be ext-real number ; ::_thesis: ( r <= p implies [.r,+infty.] \ {p} = [.r,p.[ \/ ].p,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( r <= p implies [.r,+infty.] \ {p} = [.r,p.[ \/ ].p,+infty.] ) by Th316; ::_thesis: verum
end;
theorem :: XXREAL_1:369
for r, p being ext-real number holds [.r,+infty.] \ ].p,+infty.[ = [.r,p.] \/ {+infty}
proof
let r, p be ext-real number ; ::_thesis: [.r,+infty.] \ ].p,+infty.[ = [.r,p.] \/ {+infty}
A1: r <= +infty by XXREAL_0:3;
p <= +infty by XXREAL_0:3;
hence [.r,+infty.] \ ].p,+infty.[ = [.r,p.] \/ {+infty} by A1, Th318; ::_thesis: verum
end;
theorem :: XXREAL_1:370
for r, p being ext-real number holds [.r,+infty.] \ [.p,+infty.[ = [.r,p.[ \/ {+infty}
proof
let r, p be ext-real number ; ::_thesis: [.r,+infty.] \ [.p,+infty.[ = [.r,p.[ \/ {+infty}
A1: r <= +infty by XXREAL_0:3;
p <= +infty by XXREAL_0:3;
hence [.r,+infty.] \ [.p,+infty.[ = [.r,p.[ \/ {+infty} by A1, Th320; ::_thesis: verum
end;
theorem :: XXREAL_1:371
for p being ext-real number
for r being real number holds ].r,+infty.] \ ].p,+infty.[ = ].r,p.] \/ {+infty}
proof
let p be ext-real number ; ::_thesis: for r being real number holds ].r,+infty.] \ ].p,+infty.[ = ].r,p.] \/ {+infty}
let r be real number ; ::_thesis: ].r,+infty.] \ ].p,+infty.[ = ].r,p.] \/ {+infty}
r in REAL by XREAL_0:def_1;
then r < +infty by XXREAL_0:9;
hence ].r,+infty.] \ ].p,+infty.[ = ].r,p.] \/ {+infty} by Th317, XXREAL_0:3; ::_thesis: verum
end;
theorem :: XXREAL_1:372
for p being ext-real number
for r being real number holds ].r,+infty.] \ [.p,+infty.[ = ].r,p.[ \/ {+infty}
proof
let p be ext-real number ; ::_thesis: for r being real number holds ].r,+infty.] \ [.p,+infty.[ = ].r,p.[ \/ {+infty}
let r be real number ; ::_thesis: ].r,+infty.] \ [.p,+infty.[ = ].r,p.[ \/ {+infty}
r in REAL by XREAL_0:def_1;
then r < +infty by XXREAL_0:9;
hence ].r,+infty.] \ [.p,+infty.[ = ].r,p.[ \/ {+infty} by Th319, XXREAL_0:3; ::_thesis: verum
end;
theorem :: XXREAL_1:373
for p, s being ext-real number st p <= s holds
[.p,+infty.] \ ].p,s.[ = {p} \/ [.s,+infty.]
proof
let p, s be ext-real number ; ::_thesis: ( p <= s implies [.p,+infty.] \ ].p,s.[ = {p} \/ [.s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( p <= s implies [.p,+infty.] \ ].p,s.[ = {p} \/ [.s,+infty.] ) by Th322; ::_thesis: verum
end;
theorem :: XXREAL_1:374
for p, s being ext-real number st p <= s holds
[.p,+infty.] \ ].p,s.] = {p} \/ ].s,+infty.]
proof
let p, s be ext-real number ; ::_thesis: ( p <= s implies [.p,+infty.] \ ].p,s.] = {p} \/ ].s,+infty.] )
p <= +infty by XXREAL_0:3;
hence ( p <= s implies [.p,+infty.] \ ].p,s.] = {p} \/ ].s,+infty.] ) by Th324; ::_thesis: verum
end;
theorem :: XXREAL_1:375
for p, s being ext-real number holds [.-infty,+infty.] \ ].p,s.[ = [.-infty,p.] \/ [.s,+infty.] by Th326, XXREAL_0:3;
theorem :: XXREAL_1:376
for p, s being ext-real number holds [.-infty,+infty.[ \ [.p,s.[ = [.-infty,p.[ \/ [.s,+infty.[ by Th327, XXREAL_0:3;
theorem :: XXREAL_1:377
for p, s being ext-real number holds [.-infty,+infty.] \ [.p,s.[ = [.-infty,p.[ \/ [.s,+infty.] by Th328, XXREAL_0:3;
theorem :: XXREAL_1:378
for p, s being ext-real number holds [.-infty,+infty.] \ ].p,s.] = [.-infty,p.] \/ ].s,+infty.] by Th330, XXREAL_0:3;
theorem :: XXREAL_1:379
for p, s being ext-real number holds [.-infty,+infty.[ \ [.p,s.] = [.-infty,p.[ \/ ].s,+infty.[ by Th331, XXREAL_0:3;
theorem :: XXREAL_1:380
for p, s being ext-real number holds [.-infty,+infty.] \ [.p,s.] = [.-infty,p.[ \/ ].s,+infty.] by Th332, XXREAL_0:3;
theorem :: XXREAL_1:381
for p being ext-real number
for s being real number holds ].-infty,+infty.] \ ].p,s.[ = ].-infty,p.] \/ [.s,+infty.] by Th334, XXREAL_0:3;
theorem :: XXREAL_1:382
for p being ext-real number
for s being real number holds REAL \ [.p,s.[ = ].-infty,p.[ \/ [.s,+infty.[ by Th224, Th335, XXREAL_0:3;
theorem :: XXREAL_1:383
for p being ext-real number
for s being real number holds ].-infty,+infty.] \ [.p,s.[ = ].-infty,p.[ \/ [.s,+infty.] by Th336, XXREAL_0:3;
theorem :: XXREAL_1:384
for p being ext-real number
for s being real number holds ].-infty,+infty.] \ ].p,s.] = ].-infty,p.] \/ ].s,+infty.] by Th338, XXREAL_0:3;
theorem :: XXREAL_1:385
for p, s being ext-real number holds REAL \ [.p,s.] = ].-infty,p.[ \/ ].s,+infty.[ by Th224, Th339, XXREAL_0:3;
theorem :: XXREAL_1:386
for p being ext-real number
for s being real number holds ].-infty,+infty.] \ [.p,s.] = ].-infty,p.[ \/ ].s,+infty.] by Th340, XXREAL_0:3;
theorem :: XXREAL_1:387
for p being ext-real number holds [.-infty,+infty.[ \ {p} = [.-infty,p.[ \/ ].p,+infty.[ by Th341, XXREAL_0:3;
theorem :: XXREAL_1:388
for p being ext-real number holds [.-infty,+infty.] \ {p} = [.-infty,p.[ \/ ].p,+infty.] by Th342, XXREAL_0:3;
theorem :: XXREAL_1:389
for p being ext-real number holds REAL \ {p} = ].-infty,p.[ \/ ].p,+infty.[ by Th224, Th349, XXREAL_0:3;
theorem :: XXREAL_1:390
for p being real number holds ].-infty,+infty.] \ {p} = ].-infty,p.[ \/ ].p,+infty.] by Th350, XXREAL_0:3;
theorem :: XXREAL_1:391
for r, s being ext-real number
for p being real number st r < s holds
].r,+infty.[ \ ].p,s.[ = ].r,p.] \/ [.s,+infty.[
proof
let r, s be ext-real number ; ::_thesis: for p being real number st r < s holds
].r,+infty.[ \ ].p,s.[ = ].r,p.] \/ [.s,+infty.[
let p be real number ; ::_thesis: ( r < s implies ].r,+infty.[ \ ].p,s.[ = ].r,p.] \/ [.s,+infty.[ )
p in REAL by XREAL_0:def_1;
then p < +infty by XXREAL_0:9;
hence ( r < s implies ].r,+infty.[ \ ].p,s.[ = ].r,p.] \/ [.s,+infty.[ ) by Th297; ::_thesis: verum
end;
theorem :: XXREAL_1:392
for r, s being ext-real number
for p being real number st r <= s holds
[.r,+infty.[ \ ].p,s.[ = [.r,p.] \/ [.s,+infty.[
proof
let r, s be ext-real number ; ::_thesis: for p being real number st r <= s holds
[.r,+infty.[ \ ].p,s.[ = [.r,p.] \/ [.s,+infty.[
let p be real number ; ::_thesis: ( r <= s implies [.r,+infty.[ \ ].p,s.[ = [.r,p.] \/ [.s,+infty.[ )
p in REAL by XREAL_0:def_1;
hence ( r <= s implies [.r,+infty.[ \ ].p,s.[ = [.r,p.] \/ [.s,+infty.[ ) by Th298, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:393
for r, s being ext-real number
for p being real number st r < s holds
].r,+infty.[ \ ].p,s.] = ].r,p.] \/ ].s,+infty.[
proof
let r, s be ext-real number ; ::_thesis: for p being real number st r < s holds
].r,+infty.[ \ ].p,s.] = ].r,p.] \/ ].s,+infty.[
let p be real number ; ::_thesis: ( r < s implies ].r,+infty.[ \ ].p,s.] = ].r,p.] \/ ].s,+infty.[ )
p in REAL by XREAL_0:def_1;
then p < +infty by XXREAL_0:9;
hence ( r < s implies ].r,+infty.[ \ ].p,s.] = ].r,p.] \/ ].s,+infty.[ ) by Th305; ::_thesis: verum
end;
theorem :: XXREAL_1:394
for r, s being ext-real number
for p being real number st r <= s holds
[.r,+infty.[ \ ].p,s.] = [.r,p.] \/ ].s,+infty.[
proof
let r, s be ext-real number ; ::_thesis: for p being real number st r <= s holds
[.r,+infty.[ \ ].p,s.] = [.r,p.] \/ ].s,+infty.[
let p be real number ; ::_thesis: ( r <= s implies [.r,+infty.[ \ ].p,s.] = [.r,p.] \/ ].s,+infty.[ )
p in REAL by XREAL_0:def_1;
hence ( r <= s implies [.r,+infty.[ \ ].p,s.] = [.r,p.] \/ ].s,+infty.[ ) by Th306, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:395
for s being ext-real number
for p being real number st p <= s holds
[.p,+infty.[ \ ].p,s.] = {p} \/ ].s,+infty.[
proof
let s be ext-real number ; ::_thesis: for p being real number st p <= s holds
[.p,+infty.[ \ ].p,s.] = {p} \/ ].s,+infty.[
let p be real number ; ::_thesis: ( p <= s implies [.p,+infty.[ \ ].p,s.] = {p} \/ ].s,+infty.[ )
p in REAL by XREAL_0:def_1;
hence ( p <= s implies [.p,+infty.[ \ ].p,s.] = {p} \/ ].s,+infty.[ ) by Th323, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:396
for s being ext-real number
for p being real number holds [.-infty,+infty.[ \ ].p,s.[ = [.-infty,p.] \/ [.s,+infty.[
proof
let s be ext-real number ; ::_thesis: for p being real number holds [.-infty,+infty.[ \ ].p,s.[ = [.-infty,p.] \/ [.s,+infty.[
let p be real number ; ::_thesis: [.-infty,+infty.[ \ ].p,s.[ = [.-infty,p.] \/ [.s,+infty.[
p in REAL by XREAL_0:def_1;
hence [.-infty,+infty.[ \ ].p,s.[ = [.-infty,p.] \/ [.s,+infty.[ by Th325, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:397
for s being ext-real number
for p being real number holds [.-infty,+infty.[ \ ].p,s.] = [.-infty,p.] \/ ].s,+infty.[
proof
let s be ext-real number ; ::_thesis: for p being real number holds [.-infty,+infty.[ \ ].p,s.] = [.-infty,p.] \/ ].s,+infty.[
let p be real number ; ::_thesis: [.-infty,+infty.[ \ ].p,s.] = [.-infty,p.] \/ ].s,+infty.[
p in REAL by XREAL_0:def_1;
hence [.-infty,+infty.[ \ ].p,s.] = [.-infty,p.] \/ ].s,+infty.[ by Th329, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:398
for s, p being real number holds REAL \ ].p,s.[ = ].-infty,p.] \/ [.s,+infty.[
proof
let s, p be real number ; ::_thesis: REAL \ ].p,s.[ = ].-infty,p.] \/ [.s,+infty.[
p in REAL by XREAL_0:def_1;
hence REAL \ ].p,s.[ = ].-infty,p.] \/ [.s,+infty.[ by Th224, Th333, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:399
for s, p being real number holds REAL \ ].p,s.] = ].-infty,p.] \/ ].s,+infty.[
proof
let s, p be real number ; ::_thesis: REAL \ ].p,s.] = ].-infty,p.] \/ ].s,+infty.[
p in REAL by XREAL_0:def_1;
hence REAL \ ].p,s.] = ].-infty,p.] \/ ].s,+infty.[ by Th224, Th337, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:400
for s being ext-real number
for p being real number st p <= s holds
[.p,+infty.[ \ ].p,s.[ = {p} \/ [.s,+infty.[
proof
let s be ext-real number ; ::_thesis: for p being real number st p <= s holds
[.p,+infty.[ \ ].p,s.[ = {p} \/ [.s,+infty.[
let p be real number ; ::_thesis: ( p <= s implies [.p,+infty.[ \ ].p,s.[ = {p} \/ [.s,+infty.[ )
p in REAL by XREAL_0:def_1;
hence ( p <= s implies [.p,+infty.[ \ ].p,s.[ = {p} \/ [.s,+infty.[ ) by Th321, XXREAL_0:9; ::_thesis: verum
end;
theorem Th401: :: XXREAL_1:401
for r, s being ext-real number st r < s holds
[.r,s.] \ [.r,s.[ = {s}
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies [.r,s.] \ [.r,s.[ = {s} )
[.s,s.] = {s} by Th17;
hence ( r < s implies [.r,s.] \ [.r,s.[ = {s} ) by Th184; ::_thesis: verum
end;
theorem Th402: :: XXREAL_1:402
for r, s being ext-real number st r < s holds
].r,s.] \ ].r,s.[ = {s}
proof
let r, s be ext-real number ; ::_thesis: ( r < s implies ].r,s.] \ ].r,s.[ = {s} )
[.s,s.] = {s} by Th17;
hence ( r < s implies ].r,s.] \ ].r,s.[ = {s} ) by Th188; ::_thesis: verum
end;
theorem Th403: :: XXREAL_1:403
for r, t being ext-real number st r < t holds
[.r,t.] \ ].r,t.] = {r}
proof
let r, t be ext-real number ; ::_thesis: ( r < t implies [.r,t.] \ ].r,t.] = {r} )
[.r,r.] = {r} by Th17;
hence ( r < t implies [.r,t.] \ ].r,t.] = {r} ) by Th192; ::_thesis: verum
end;
theorem Th404: :: XXREAL_1:404
for r, t being ext-real number st r < t holds
[.r,t.[ \ ].r,t.[ = {r}
proof
let r, t be ext-real number ; ::_thesis: ( r < t implies [.r,t.[ \ ].r,t.[ = {r} )
[.r,r.] = {r} by Th17;
hence ( r < t implies [.r,t.[ \ ].r,t.[ = {r} ) by Th196; ::_thesis: verum
end;
theorem :: XXREAL_1:405
for s being real number holds [.-infty,s.] \ [.-infty,s.[ = {s}
proof
let s be real number ; ::_thesis: [.-infty,s.] \ [.-infty,s.[ = {s}
s in REAL by XREAL_0:def_1;
hence [.-infty,s.] \ [.-infty,s.[ = {s} by Th401, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:406
for s being real number holds ].-infty,s.] \ ].-infty,s.[ = {s}
proof
let s be real number ; ::_thesis: ].-infty,s.] \ ].-infty,s.[ = {s}
s in REAL by XREAL_0:def_1;
hence ].-infty,s.] \ ].-infty,s.[ = {s} by Th402, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:407
for s being real number holds [.-infty,s.] \ ].-infty,s.] = {-infty}
proof
let s be real number ; ::_thesis: [.-infty,s.] \ ].-infty,s.] = {-infty}
s in REAL by XREAL_0:def_1;
hence [.-infty,s.] \ ].-infty,s.] = {-infty} by Th403, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:408
for s being real number holds [.-infty,s.[ \ ].-infty,s.[ = {-infty}
proof
let s be real number ; ::_thesis: [.-infty,s.[ \ ].-infty,s.[ = {-infty}
s in REAL by XREAL_0:def_1;
hence [.-infty,s.[ \ ].-infty,s.[ = {-infty} by Th404, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:409
for s being real number holds [.s,+infty.] \ [.s,+infty.[ = {+infty}
proof
let s be real number ; ::_thesis: [.s,+infty.] \ [.s,+infty.[ = {+infty}
s in REAL by XREAL_0:def_1;
hence [.s,+infty.] \ [.s,+infty.[ = {+infty} by Th401, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:410
for s being real number holds ].s,+infty.] \ ].s,+infty.[ = {+infty}
proof
let s be real number ; ::_thesis: ].s,+infty.] \ ].s,+infty.[ = {+infty}
s in REAL by XREAL_0:def_1;
hence ].s,+infty.] \ ].s,+infty.[ = {+infty} by Th402, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:411
for s being real number holds [.s,+infty.] \ ].s,+infty.] = {s}
proof
let s be real number ; ::_thesis: [.s,+infty.] \ ].s,+infty.] = {s}
s in REAL by XREAL_0:def_1;
hence [.s,+infty.] \ ].s,+infty.] = {s} by Th403, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:412
for s being real number holds [.s,+infty.[ \ ].s,+infty.[ = {s}
proof
let s be real number ; ::_thesis: [.s,+infty.[ \ ].s,+infty.[ = {s}
s in REAL by XREAL_0:def_1;
hence [.s,+infty.[ \ ].s,+infty.[ = {s} by Th404, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:413
for r, s, t being ext-real number st r <= s & s < t holds
[.r,s.] \/ [.s,t.[ = [.r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s < t implies [.r,s.] \/ [.s,t.[ = [.r,t.[ )
assume that
A1: r <= s and
A2: s < t ; ::_thesis: [.r,s.] \/ [.s,t.[ = [.r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in [.r,s.] \/ [.s,t.[ or p in [.r,t.[ ) & ( not p in [.r,t.[ or p in [.r,s.] \/ [.s,t.[ ) )
thus ( p in [.r,s.] \/ [.s,t.[ implies p in [.r,t.[ ) ::_thesis: ( not p in [.r,t.[ or p in [.r,s.] \/ [.s,t.[ )
proof
assume p in [.r,s.] \/ [.s,t.[ ; ::_thesis: p in [.r,t.[
then ( p in [.r,s.] or p in [.s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r <= p & p <= s ) or ( s <= p & p < t ) ) by Th1, Th3;
then A4: r <= p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in [.r,t.[ by A4, Th3; ::_thesis: verum
end;
assume p in [.r,t.[ ; ::_thesis: p in [.r,s.] \/ [.s,t.[
then ( ( r <= p & p <= s ) or ( s <= p & p < t ) ) by Th3;
then ( p in [.r,s.] or p in [.s,t.[ ) by Th1, Th3;
hence p in [.r,s.] \/ [.s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:414
for r, s, t being ext-real number st r < s & s <= t holds
].r,s.] \/ [.s,t.] = ].r,t.]
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s <= t implies ].r,s.] \/ [.s,t.] = ].r,t.] )
assume that
A1: r < s and
A2: s <= t ; ::_thesis: ].r,s.] \/ [.s,t.] = ].r,t.]
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.] \/ [.s,t.] or p in ].r,t.] ) & ( not p in ].r,t.] or p in ].r,s.] \/ [.s,t.] ) )
thus ( p in ].r,s.] \/ [.s,t.] implies p in ].r,t.] ) ::_thesis: ( not p in ].r,t.] or p in ].r,s.] \/ [.s,t.] )
proof
assume p in ].r,s.] \/ [.s,t.] ; ::_thesis: p in ].r,t.]
then ( p in ].r,s.] or p in [.s,t.] ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p <= s ) or ( s <= p & p <= t ) ) by Th1, Th2;
then A4: r < p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in ].r,t.] by A4, Th2; ::_thesis: verum
end;
assume p in ].r,t.] ; ::_thesis: p in ].r,s.] \/ [.s,t.]
then ( ( r < p & p <= s ) or ( s <= p & p <= t ) ) by Th2;
then ( p in ].r,s.] or p in [.s,t.] ) by Th1, Th2;
hence p in ].r,s.] \/ [.s,t.] by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:415
for r, s, t being ext-real number st r < s & s < t holds
].r,s.] \/ [.s,t.[ = ].r,t.[
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s < t implies ].r,s.] \/ [.s,t.[ = ].r,t.[ )
assume that
A1: r < s and
A2: s < t ; ::_thesis: ].r,s.] \/ [.s,t.[ = ].r,t.[
let p be ext-real number ; :: according to MEMBERED:def_14 ::_thesis: ( ( not p in ].r,s.] \/ [.s,t.[ or p in ].r,t.[ ) & ( not p in ].r,t.[ or p in ].r,s.] \/ [.s,t.[ ) )
thus ( p in ].r,s.] \/ [.s,t.[ implies p in ].r,t.[ ) ::_thesis: ( not p in ].r,t.[ or p in ].r,s.] \/ [.s,t.[ )
proof
assume p in ].r,s.] \/ [.s,t.[ ; ::_thesis: p in ].r,t.[
then ( p in ].r,s.] or p in [.s,t.[ ) by XBOOLE_0:def_3;
then A3: ( ( r < p & p <= s ) or ( s <= p & p < t ) ) by Th2, Th3;
then A4: r < p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in ].r,t.[ by A4, Th4; ::_thesis: verum
end;
assume p in ].r,t.[ ; ::_thesis: p in ].r,s.] \/ [.s,t.[
then ( ( r < p & p <= s ) or ( s <= p & p < t ) ) by Th4;
then ( p in ].r,s.] or p in [.s,t.[ ) by Th2, Th3;
hence p in ].r,s.] \/ [.s,t.[ by XBOOLE_0:def_3; ::_thesis: verum
end;
theorem :: XXREAL_1:416
for r, s, t being ext-real number st r <= s & s < t holds
[.r,s.] /\ [.s,t.[ = {s}
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s < t implies [.r,s.] /\ [.s,t.[ = {s} )
assume that
A1: r <= s and
A2: s < t ; ::_thesis: [.r,s.] /\ [.s,t.[ = {s}
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_[.r,s.]_/\_[.s,t.[_implies_x_=_s_)_&_(_x_=_s_implies_x_in_[.r,s.]_/\_[.s,t.[_)_)
let x be set ; ::_thesis: ( ( x in [.r,s.] /\ [.s,t.[ implies x = s ) & ( x = s implies x in [.r,s.] /\ [.s,t.[ ) )
hereby ::_thesis: ( x = s implies x in [.r,s.] /\ [.s,t.[ )
assume A3: x in [.r,s.] /\ [.s,t.[ ; ::_thesis: x = s
then reconsider p = x as ext-real number ;
A4: p in [.r,s.] by A3, XBOOLE_0:def_4;
p in [.s,t.[ by A3, XBOOLE_0:def_4;
then A5: s <= p by Th3;
p <= s by A4, Th1;
hence x = s by A5, XXREAL_0:1; ::_thesis: verum
end;
assume A6: x = s ; ::_thesis: x in [.r,s.] /\ [.s,t.[
A7: s in [.r,s.] by A1, Th1;
s in [.s,t.[ by A2, Th3;
hence x in [.r,s.] /\ [.s,t.[ by A6, A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence [.r,s.] /\ [.s,t.[ = {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:417
for r, s, t being ext-real number st r < s & s <= t holds
].r,s.] /\ [.s,t.] = {s}
proof
let r, s, t be ext-real number ; ::_thesis: ( r < s & s <= t implies ].r,s.] /\ [.s,t.] = {s} )
assume that
A1: r < s and
A2: s <= t ; ::_thesis: ].r,s.] /\ [.s,t.] = {s}
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_].r,s.]_/\_[.s,t.]_implies_x_=_s_)_&_(_x_=_s_implies_x_in_].r,s.]_/\_[.s,t.]_)_)
let x be set ; ::_thesis: ( ( x in ].r,s.] /\ [.s,t.] implies x = s ) & ( x = s implies x in ].r,s.] /\ [.s,t.] ) )
hereby ::_thesis: ( x = s implies x in ].r,s.] /\ [.s,t.] )
assume A3: x in ].r,s.] /\ [.s,t.] ; ::_thesis: x = s
then reconsider p = x as ext-real number ;
A4: p in ].r,s.] by A3, XBOOLE_0:def_4;
p in [.s,t.] by A3, XBOOLE_0:def_4;
then A5: s <= p by Th1;
p <= s by A4, Th2;
hence x = s by A5, XXREAL_0:1; ::_thesis: verum
end;
assume A6: x = s ; ::_thesis: x in ].r,s.] /\ [.s,t.]
A7: s in ].r,s.] by A1, Th2;
s in [.s,t.] by A2, Th1;
hence x in ].r,s.] /\ [.s,t.] by A6, A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence ].r,s.] /\ [.s,t.] = {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:418
for r, s, t being ext-real number st r <= s & s <= t holds
[.r,s.] /\ [.s,t.] = {s}
proof
let r, s, t be ext-real number ; ::_thesis: ( r <= s & s <= t implies [.r,s.] /\ [.s,t.] = {s} )
assume that
A1: r <= s and
A2: s <= t ; ::_thesis: [.r,s.] /\ [.s,t.] = {s}
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_[.r,s.]_/\_[.s,t.]_implies_x_=_s_)_&_(_x_=_s_implies_x_in_[.r,s.]_/\_[.s,t.]_)_)
let x be set ; ::_thesis: ( ( x in [.r,s.] /\ [.s,t.] implies x = s ) & ( x = s implies x in [.r,s.] /\ [.s,t.] ) )
hereby ::_thesis: ( x = s implies x in [.r,s.] /\ [.s,t.] )
assume A3: x in [.r,s.] /\ [.s,t.] ; ::_thesis: x = s
then reconsider p = x as ext-real number ;
A4: p in [.r,s.] by A3, XBOOLE_0:def_4;
p in [.s,t.] by A3, XBOOLE_0:def_4;
then A5: s <= p by Th1;
p <= s by A4, Th1;
hence x = s by A5, XXREAL_0:1; ::_thesis: verum
end;
assume A6: x = s ; ::_thesis: x in [.r,s.] /\ [.s,t.]
A7: s in [.r,s.] by A1, Th1;
s in [.s,t.] by A2, Th1;
hence x in [.r,s.] /\ [.s,t.] by A6, A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence [.r,s.] /\ [.s,t.] = {s} by TARSKI:def_1; ::_thesis: verum
end;
theorem :: XXREAL_1:419
for s being ext-real number holds [.-infty,s.] = ].-infty,s.[ \/ {-infty,s} by Th128, XXREAL_0:5;
theorem :: XXREAL_1:420
for s being ext-real number holds [.-infty,s.] = [.-infty,s.[ \/ {s} by Th129, XXREAL_0:5;
theorem :: XXREAL_1:421
for s being ext-real number holds [.-infty,s.] = {-infty} \/ ].-infty,s.] by Th130, XXREAL_0:5;
theorem :: XXREAL_1:422
for s being real number holds [.-infty,s.[ = {-infty} \/ ].-infty,s.[
proof
let s be real number ; ::_thesis: [.-infty,s.[ = {-infty} \/ ].-infty,s.[
s in REAL by XREAL_0:def_1;
hence [.-infty,s.[ = {-infty} \/ ].-infty,s.[ by Th131, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:423
for s being real number holds ].-infty,s.] = ].-infty,s.[ \/ {s}
proof
let s be real number ; ::_thesis: ].-infty,s.] = ].-infty,s.[ \/ {s}
s in REAL by XREAL_0:def_1;
hence ].-infty,s.] = ].-infty,s.[ \/ {s} by Th132, XXREAL_0:12; ::_thesis: verum
end;
theorem :: XXREAL_1:424
for r being ext-real number holds [.r,+infty.] = ].r,+infty.[ \/ {r,+infty} by Th128, XXREAL_0:3;
theorem :: XXREAL_1:425
for r being ext-real number holds [.r,+infty.] = [.r,+infty.[ \/ {+infty} by Th129, XXREAL_0:3;
theorem :: XXREAL_1:426
for r being ext-real number holds [.r,+infty.] = {r} \/ ].r,+infty.] by Th130, XXREAL_0:3;
theorem :: XXREAL_1:427
for r being real number holds [.r,+infty.[ = {r} \/ ].r,+infty.[
proof
let r be real number ; ::_thesis: [.r,+infty.[ = {r} \/ ].r,+infty.[
r in REAL by XREAL_0:def_1;
hence [.r,+infty.[ = {r} \/ ].r,+infty.[ by Th131, XXREAL_0:9; ::_thesis: verum
end;
theorem :: XXREAL_1:428
for r being real number holds ].r,+infty.] = ].r,+infty.[ \/ {+infty}
proof
let r be real number ; ::_thesis: ].r,+infty.] = ].r,+infty.[ \/ {+infty}
r in REAL by XREAL_0:def_1;
hence ].r,+infty.] = ].r,+infty.[ \/ {+infty} by Th132, XXREAL_0:9; ::_thesis: verum
end;