:: RCOMP_3 semantic presentation

begin

registration
let X be non empty set ;
cluster [#] X ->  non empty ;
coherence
 not [#] X is empty ;
end;

registration
cluster  ->  real-membered for SubSpace of RealSpace ;
coherence
 for b1 being SubSpace of RealSpace holds b1 is real-membered
proof
let T be SubSpace of RealSpace ; ::_thesis: T is real-membered
 the carrier of T is Subset of RealSpace by TOPMETR:def_1;
hence  the carrier of T is real-membered ; :: according to TOPMETR:def_8 ::_thesis: verum
end;
end;

theorem Th1: :: RCOMP_3:1
for X being non empty real-membered bounded_below set
for Y being closed Subset of REAL st X c= Y holds
lower_bound X in Y
proof
let X be non empty real-membered bounded_below set ; ::_thesis: for Y being closed Subset of REAL st X c= Y holds
lower_bound X in Y
let Y be closed Subset of REAL; ::_thesis: ( X c= Y implies lower_bound X in Y )
assume A1: X c= Y ; ::_thesis: lower_bound X in Y
reconsider X = X as non empty bounded_below Subset of REAL by MEMBERED:3;
A2: ( lower_bound X = lower_bound (Cl X) & lower_bound (Cl X) in Cl X ) by RCOMP_1:13, TOPREAL6:68;
 Cl X c= Y by A1, MEASURE6:57;
hence  lower_bound X in Y by A2; ::_thesis: verum
end;

theorem Th2: :: RCOMP_3:2
for X being non empty real-membered bounded_above set
for Y being closed Subset of REAL st X c= Y holds
upper_bound X in Y
proof
let X be non empty real-membered bounded_above set ; ::_thesis: for Y being closed Subset of REAL st X c= Y holds
upper_bound X in Y
let Y be closed Subset of REAL; ::_thesis: ( X c= Y implies upper_bound X in Y )
assume A1: X c= Y ; ::_thesis: upper_bound X in Y
reconsider X = X as non empty bounded_above Subset of REAL by MEMBERED:3;
A2: ( upper_bound X = upper_bound (Cl X) & upper_bound (Cl X) in Cl X ) by RCOMP_1:12, TOPREAL6:69;
 Cl X c= Y by A1, MEASURE6:57;
hence  upper_bound X in Y by A2; ::_thesis: verum
end;

theorem Th3: :: RCOMP_3:3
for X, Y being Subset of REAL holds Cl (X \/ Y) = (Cl X) \/ (Cl Y)
proof
let X, Y be Subset of REAL; ::_thesis: Cl (X \/ Y) = (Cl X) \/ (Cl Y)
reconsider A = X, B = Y as Subset of R^1 by TOPMETR:17;
thus  Cl (X \/ Y) = Cl (A \/ B) by JORDAN5A:24
.= (Cl A) \/ (Cl B) by PRE_TOPC:20
.= (Cl X) \/ (Cl B) by JORDAN5A:24
.= (Cl X) \/ (Cl Y) by JORDAN5A:24 ; ::_thesis: verum
end;

begin


registration
let r be real number ;
let s be ext-real number ;
cluster[.r,s.[ ->  bounded_below ;
coherence
 [.r,s.[ is bounded_below
proof
take  r ; :: according to XXREAL_2:def_9 ::_thesis: r is LowerBound of [.r,s.[
let x be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not x in [.r,s.[ or r <= x )
thus  ( not x in [.r,s.[ or r <= x ) by XXREAL_1:3; ::_thesis: verum
end;
cluster].s,r.] ->  bounded_above ;
coherence
 ].s,r.] is bounded_above
proof
take  r ; :: according to XXREAL_2:def_10 ::_thesis: r is UpperBound of ].s,r.]
let x be ext-real number ; :: according to XXREAL_2:def_1 ::_thesis: ( not x in ].s,r.] or x <= r )
thus  ( not x in ].s,r.] or x <= r ) by XXREAL_1:2; ::_thesis: verum
end;
end;

registration
let r, s be real number ;
cluster[.r,s.[ ->  real-bounded ;
coherence
 [.r,s.[ is real-bounded
proof
 [.r,s.[ is bounded_above
proof
take  s ; :: according to XXREAL_2:def_10 ::_thesis: s is UpperBound of [.r,s.[
let x be ext-real number ; :: according to XXREAL_2:def_1 ::_thesis: ( not x in [.r,s.[ or x <= s )
thus  ( not x in [.r,s.[ or x <= s ) by XXREAL_1:3; ::_thesis: verum
end;
hence  [.r,s.[ is real-bounded ; ::_thesis: verum
end;
cluster].r,s.] ->  real-bounded ;
coherence
 ].r,s.] is real-bounded
proof
 ].r,s.] is bounded_below
proof
take  r ; :: according to XXREAL_2:def_9 ::_thesis: r is LowerBound of ].r,s.]
let x be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not x in ].r,s.] or r <= x )
thus  ( not x in ].r,s.] or r <= x ) by XXREAL_1:2; ::_thesis: verum
end;
hence  ].r,s.] is real-bounded ; ::_thesis: verum
end;
cluster].r,s.[ ->  real-bounded ;
coherence
 ].r,s.[ is real-bounded
proof
A1: ].r,s.[ is bounded_above
proof
take  s ; :: according to XXREAL_2:def_10 ::_thesis: s is UpperBound of ].r,s.[
let x be ext-real number ; :: according to XXREAL_2:def_1 ::_thesis: ( not x in ].r,s.[ or x <= s )
thus  ( not x in ].r,s.[ or x <= s ) by XXREAL_1:4; ::_thesis: verum
end;
 ].r,s.[ is bounded_below
proof
take  r ; :: according to XXREAL_2:def_9 ::_thesis: r is LowerBound of ].r,s.[
let x be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not x in ].r,s.[ or r <= x )
thus  ( not x in ].r,s.[ or r <= x ) by XXREAL_1:4; ::_thesis: verum
end;
hence  ].r,s.[ is real-bounded by A1; ::_thesis: verum
end;
end;

registration
cluster non empty complex-membered ext-real-membered real-membered open real-bounded interval for Element of K32(REAL);
existence
 ex b1 being Subset of REAL st
( b1 is open & b1 is real-bounded & b1 is interval & not b1 is empty )
proof
take  ].0,1.[ ; ::_thesis: ( ].0,1.[ is open & ].0,1.[ is real-bounded & ].0,1.[ is interval & not ].0,1.[ is empty )
thus  ( ].0,1.[ is open & ].0,1.[ is real-bounded & ].0,1.[ is interval & not ].0,1.[ is empty ) ; ::_thesis: verum
end;
end;

theorem Th4: :: RCOMP_3:4
for r, s being real number st r < s holds
lower_bound [.r,s.[ = r
proof
let r, s be real number ; ::_thesis: ( r < s implies lower_bound [.r,s.[ = r )
set X = [.r,s.[;
assume A1: r < s ; ::_thesis: lower_bound [.r,s.[ = r
A2: for a being real number st a in [.r,s.[ holds
r <= a by XXREAL_1:3;
A3: (r + s) / 2 < s by A1, XREAL_1:226;
A4: r < (r + s) / 2 by A1, XREAL_1:226;
A5: for b being real number st 0 < b holds
ex a being real number st
( a in [.r,s.[ & a < r + b )
proof
let b be real number ; ::_thesis: ( 0 < b implies ex a being real number st
( a in [.r,s.[ & a < r + b ) )
assume that
A6: 0 < b and
A7: for a being real number st a in [.r,s.[ holds
a >= r + b ; ::_thesis: contradiction
percases ( r + b > s or r + b <= s ) ;
suppose r + b > s ; ::_thesis: contradiction
then A8: (r + s) / 2 < r + b by A3, XXREAL_0:2;
 (r + s) / 2 in [.r,s.[ by A4, A3, XXREAL_1:3;
hence  contradiction by A7, A8; ::_thesis: verum
end;
supposeA9: r + b <= s ; ::_thesis: contradiction
A10: r < r + b by A6, XREAL_1:29;
then  (r + (r + b)) / 2 < r + b by XREAL_1:226;
then A11: (r + (r + b)) / 2 < s by A9, XXREAL_0:2;
 r < (r + (r + b)) / 2 by A10, XREAL_1:226;
then  (r + (r + b)) / 2 in [.r,s.[ by A11, XXREAL_1:3;
hence  contradiction by A7, A10, XREAL_1:226; ::_thesis: verum
end;
end;
end;
 not [.r,s.[ is empty by A1, XXREAL_1:3;
hence  lower_bound [.r,s.[ = r by A2, A5, SEQ_4:def_2; ::_thesis: verum
end;

theorem Th5: :: RCOMP_3:5
for r, s being real number st r < s holds
upper_bound [.r,s.[ = s
proof
let r, s be real number ; ::_thesis: ( r < s implies upper_bound [.r,s.[ = s )
set X = [.r,s.[;
assume A1: r < s ; ::_thesis: upper_bound [.r,s.[ = s
A2: for a being real number st a in [.r,s.[ holds
a <= s by XXREAL_1:3;
A3: r < (r + s) / 2 by A1, XREAL_1:226;
A4: (r + s) / 2 < s by A1, XREAL_1:226;
A5: for b being real number st 0 < b holds
ex a being real number st
( a in [.r,s.[ & s - b < a )
proof
let b be real number ; ::_thesis: ( 0 < b implies ex a being real number st
( a in [.r,s.[ & s - b < a ) )
assume that
A6: 0 < b and
A7: for a being real number st a in [.r,s.[ holds
a <= s - b ; ::_thesis: contradiction
percases ( s - b <= r or s - b > r ) ;
suppose s - b <= r ; ::_thesis: contradiction
then A8: (r + s) / 2 > s - b by A3, XXREAL_0:2;
 (r + s) / 2 in [.r,s.[ by A3, A4, XXREAL_1:3;
hence  contradiction by A7, A8; ::_thesis: verum
end;
supposeA9: s - b > r ; ::_thesis: contradiction
A10: s - b < s - 0 by A6, XREAL_1:15;
then  s - b < (s + (s - b)) / 2 by XREAL_1:226;
then A11: r < (s + (s - b)) / 2 by A9, XXREAL_0:2;
 (s + (s - b)) / 2 < s by A10, XREAL_1:226;
then  (s + (s - b)) / 2 in [.r,s.[ by A11, XXREAL_1:3;
hence  contradiction by A7, A10, XREAL_1:226; ::_thesis: verum
end;
end;
end;
 not [.r,s.[ is empty by A1, XXREAL_1:3;
hence  upper_bound [.r,s.[ = s by A2, A5, SEQ_4:def_1; ::_thesis: verum
end;

theorem Th6: :: RCOMP_3:6
for r, s being real number st r < s holds
lower_bound ].r,s.] = r
proof
let r, s be real number ; ::_thesis: ( r < s implies lower_bound ].r,s.] = r )
set X = ].r,s.];
assume A1: r < s ; ::_thesis: lower_bound ].r,s.] = r
A2: for a being real number st a in ].r,s.] holds
r <= a by XXREAL_1:2;
A3: (r + s) / 2 < s by A1, XREAL_1:226;
A4: r < (r + s) / 2 by A1, XREAL_1:226;
A5: for b being real number st 0 < b holds
ex a being real number st
( a in ].r,s.] & a < r + b )
proof
let b be real number ; ::_thesis: ( 0 < b implies ex a being real number st
( a in ].r,s.] & a < r + b ) )
assume that
A6: 0 < b and
A7: for a being real number st a in ].r,s.] holds
a >= r + b ; ::_thesis: contradiction
percases ( r + b > s or r + b <= s ) ;
suppose r + b > s ; ::_thesis: contradiction
then A8: (r + s) / 2 < r + b by A3, XXREAL_0:2;
 (r + s) / 2 in ].r,s.] by A4, A3, XXREAL_1:2;
hence  contradiction by A7, A8; ::_thesis: verum
end;
supposeA9: r + b <= s ; ::_thesis: contradiction
A10: r < r + b by A6, XREAL_1:29;
then  (r + (r + b)) / 2 < r + b by XREAL_1:226;
then A11: (r + (r + b)) / 2 < s by A9, XXREAL_0:2;
 r < (r + (r + b)) / 2 by A10, XREAL_1:226;
then  (r + (r + b)) / 2 in ].r,s.] by A11, XXREAL_1:2;
hence  contradiction by A7, A10, XREAL_1:226; ::_thesis: verum
end;
end;
end;
 not ].r,s.] is empty by A1, XXREAL_1:2;
hence  lower_bound ].r,s.] = r by A2, A5, SEQ_4:def_2; ::_thesis: verum
end;

theorem Th7: :: RCOMP_3:7
for r, s being real number st r < s holds
upper_bound ].r,s.] = s
proof
let r, s be real number ; ::_thesis: ( r < s implies upper_bound ].r,s.] = s )
set X = ].r,s.];
assume A1: r < s ; ::_thesis: upper_bound ].r,s.] = s
A2: for a being real number st a in ].r,s.] holds
a <= s by XXREAL_1:2;
A3: r < (r + s) / 2 by A1, XREAL_1:226;
A4: (r + s) / 2 < s by A1, XREAL_1:226;
A5: for b being real number st 0 < b holds
ex a being real number st
( a in ].r,s.] & s - b < a )
proof
let b be real number ; ::_thesis: ( 0 < b implies ex a being real number st
( a in ].r,s.] & s - b < a ) )
assume that
A6: 0 < b and
A7: for a being real number st a in ].r,s.] holds
a <= s - b ; ::_thesis: contradiction
percases ( s - b <= r or s - b > r ) ;
suppose s - b <= r ; ::_thesis: contradiction
then A8: (r + s) / 2 > s - b by A3, XXREAL_0:2;
 (r + s) / 2 in ].r,s.] by A3, A4, XXREAL_1:2;
hence  contradiction by A7, A8; ::_thesis: verum
end;
supposeA9: s - b > r ; ::_thesis: contradiction
A10: s - b < s - 0 by A6, XREAL_1:15;
then  s - b < (s + (s - b)) / 2 by XREAL_1:226;
then A11: r < (s + (s - b)) / 2 by A9, XXREAL_0:2;
 (s + (s - b)) / 2 < s by A10, XREAL_1:226;
then  (s + (s - b)) / 2 in ].r,s.] by A11, XXREAL_1:2;
hence  contradiction by A7, A10, XREAL_1:226; ::_thesis: verum
end;
end;
end;
 not ].r,s.] is empty by A1, XXREAL_1:2;
hence  upper_bound ].r,s.] = s by A2, A5, SEQ_4:def_1; ::_thesis: verum
end;

begin

theorem Th8: :: RCOMP_3:8
for a, b being real number st a <= b holds
[.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b}
proof
let a, b be real number ; ::_thesis: ( a <= b implies [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} )
set A = left_closed_halfline a;
set B = right_closed_halfline b;
assume  a <= b ; ::_thesis: [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b}
then A1: ( a in [.a,b.] & b in [.a,b.] ) by XXREAL_1:1;
thus  [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) c= {a,b} :: according to XBOOLE_0:def_10 ::_thesis: {a,b} c= [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) or x in {a,b} )
assume A2: x in [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) ; ::_thesis: x in {a,b}
then reconsider x = x as Real ;
 x in (left_closed_halfline a) \/ (right_closed_halfline b) by A2, XBOOLE_0:def_4;
then  ( x in left_closed_halfline a or x in right_closed_halfline b ) by XBOOLE_0:def_3;
then A3: ( x <= a or x >= b ) by XXREAL_1:234, XXREAL_1:236;
 x in [.a,b.] by A2, XBOOLE_0:def_4;
then  ( a <= x & x <= b ) by XXREAL_1:1;
then  ( x = a or x = b ) by A3, XXREAL_0:1;
hence  x in {a,b} by TARSKI:def_2; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {a,b} or x in [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) )
 a in left_closed_halfline a by XXREAL_1:234;
then A4: a in (left_closed_halfline a) \/ (right_closed_halfline b) by XBOOLE_0:def_3;
 b in right_closed_halfline b by XXREAL_1:236;
then A5: b in (left_closed_halfline a) \/ (right_closed_halfline b) by XBOOLE_0:def_3;
assume  x in {a,b} ; ::_thesis: x in [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b))
then  ( x = a or x = b ) by TARSKI:def_2;
hence  x in [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) by A1, A4, A5, XBOOLE_0:def_4; ::_thesis: verum
end;

Lm1: now__::_thesis:_for_a_being_real_number_holds_not_left_open_halfline_a_is_bounded_below
let a be real number ; ::_thesis: not left_open_halfline a is bounded_below
assume  left_open_halfline a is bounded_below ; ::_thesis: contradiction
then consider b being real number  such that
A1: b is LowerBound of left_open_halfline a by XXREAL_2:def_9;
A2: for r being real number st r in left_open_halfline a holds
b <= r by A1, XXREAL_2:def_2;
A3: a - 1 < a - 0 by XREAL_1:15;
then  a - 1 in left_open_halfline a by XXREAL_1:233;
then  b - 1 < (a - 1) - 0 by A2, XREAL_1:15;
then  b - 1 < a by A3, XXREAL_0:2;
then  b - 1 in left_open_halfline a by XXREAL_1:233;
then  b - 0 <= b - 1 by A1, XXREAL_2:def_2;
hence  contradiction by XREAL_1:15; ::_thesis: verum
end;

Lm2: now__::_thesis:_for_a_being_real_number_holds_not_right_open_halfline_a_is_bounded_above
let a be real number ; ::_thesis: not right_open_halfline a is bounded_above
assume  right_open_halfline a is bounded_above ; ::_thesis: contradiction
then consider b being real number  such that
A1: b is UpperBound of right_open_halfline a by XXREAL_2:def_10;
A2: a + 0 < a + 1 by XREAL_1:6;
then  a + 1 in right_open_halfline a by XXREAL_1:235;
then  a + 1 <= b by A1, XXREAL_2:def_1;
then  (a + 1) + 0 <= b + 1 by XREAL_1:7;
then  a < b + 1 by A2, XXREAL_0:2;
then  b + 1 in right_open_halfline a by XXREAL_1:235;
then  b + 1 <= b + 0 by A1, XXREAL_2:def_1;
hence  contradiction by XREAL_1:6; ::_thesis: verum
end;

registration
let a be real number ;
cluster left_closed_halfline a ->  non bounded_below bounded_above interval ;
coherence
 ( not left_closed_halfline a is bounded_below & left_closed_halfline a is bounded_above & left_closed_halfline a is interval )
proof
set A = left_closed_halfline a;
 not left_open_halfline a is bounded_below by Lm1;
hence  not left_closed_halfline a is bounded_below by XXREAL_1:21, XXREAL_2:44; ::_thesis: ( left_closed_halfline a is bounded_above & left_closed_halfline a is interval )
thus  left_closed_halfline a is bounded_above ; ::_thesis: left_closed_halfline a is interval
let r, s be ext-real number ; :: according to XXREAL_2:def_12 ::_thesis: ( not r in left_closed_halfline a or not s in left_closed_halfline a or [.r,s.] c= left_closed_halfline a )
assume A1: ( r in left_closed_halfline a & s in left_closed_halfline a ) ; ::_thesis: [.r,s.] c= left_closed_halfline a
then reconsider rr = r, ss = s as Real ;
A2: s <= a by A1, XXREAL_1:234;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.r,s.] or x in left_closed_halfline a )
assume A3: x in [.r,s.] ; ::_thesis: x in left_closed_halfline a
then  x in [.rr,ss.] ;
then reconsider x = x as Real ;
 x <= s by A3, XXREAL_1:1;
then  x <= a by A2, XXREAL_0:2;
hence  x in left_closed_halfline a by XXREAL_1:234; ::_thesis: verum
end;
cluster left_open_halfline a ->  non bounded_below bounded_above interval ;
coherence
 ( not left_open_halfline a is bounded_below & left_open_halfline a is bounded_above & left_open_halfline a is interval )
proof
set A = left_open_halfline a;
thus  not left_open_halfline a is bounded_below by Lm1; ::_thesis: ( left_open_halfline a is bounded_above & left_open_halfline a is interval )
thus  left_open_halfline a is bounded_above ::_thesis: left_open_halfline a is interval
proof
take  a ; :: according to XXREAL_2:def_10 ::_thesis: a is UpperBound of left_open_halfline a
let x be ext-real number ; :: according to XXREAL_2:def_1 ::_thesis: ( not x in left_open_halfline a or x <= a )
thus  ( not x in left_open_halfline a or x <= a ) by XXREAL_1:233; ::_thesis: verum
end;
let r, s be ext-real number ; :: according to XXREAL_2:def_12 ::_thesis: ( not r in left_open_halfline a or not s in left_open_halfline a or [.r,s.] c= left_open_halfline a )
assume A4: r in left_open_halfline a ; ::_thesis: ( not s in left_open_halfline a or [.r,s.] c= left_open_halfline a )
assume A5: s in left_open_halfline a ; ::_thesis: [.r,s.] c= left_open_halfline a
then A6: s < a by XXREAL_1:233;
reconsider rr = r, ss = s as Real by A4, A5;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.r,s.] or x in left_open_halfline a )
assume A7: x in [.r,s.] ; ::_thesis: x in left_open_halfline a
then  x in [.rr,ss.] ;
then reconsider x = x as Real ;
 x <= s by A7, XXREAL_1:1;
then  x < a by A6, XXREAL_0:2;
hence  x in left_open_halfline a by XXREAL_1:233; ::_thesis: verum
end;
cluster right_closed_halfline a ->  bounded_below non bounded_above interval ;
coherence
 ( right_closed_halfline a is bounded_below & not right_closed_halfline a is bounded_above & right_closed_halfline a is interval )
proof
set A = right_closed_halfline a;
thus  right_closed_halfline a is bounded_below ; ::_thesis: ( not right_closed_halfline a is bounded_above & right_closed_halfline a is interval )
 not right_open_halfline a is bounded_above by Lm2;
hence  not right_closed_halfline a is bounded_above by XXREAL_1:22, XXREAL_2:43; ::_thesis: right_closed_halfline a is interval
let r, s be ext-real number ; :: according to XXREAL_2:def_12 ::_thesis: ( not r in right_closed_halfline a or not s in right_closed_halfline a or [.r,s.] c= right_closed_halfline a )
assume A8: r in right_closed_halfline a ; ::_thesis: ( not s in right_closed_halfline a or [.r,s.] c= right_closed_halfline a )
then A9: a <= r by XXREAL_1:236;
assume  s in right_closed_halfline a ; ::_thesis: [.r,s.] c= right_closed_halfline a
then reconsider rr = r, ss = s as Real by A8;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.r,s.] or x in right_closed_halfline a )
assume A10: x in [.r,s.] ; ::_thesis: x in right_closed_halfline a
then  x in [.rr,ss.] ;
then reconsider x = x as Real ;
 r <= x by A10, XXREAL_1:1;
then  a <= x by A9, XXREAL_0:2;
hence  x in right_closed_halfline a by XXREAL_1:236; ::_thesis: verum
end;
cluster right_open_halfline a ->  bounded_below non bounded_above interval ;
coherence
 ( right_open_halfline a is bounded_below & not right_open_halfline a is bounded_above & right_open_halfline a is interval )
proof
set A = right_open_halfline a;
thus  right_open_halfline a is bounded_below ::_thesis: ( not right_open_halfline a is bounded_above & right_open_halfline a is interval )
proof
take  a ; :: according to XXREAL_2:def_9 ::_thesis: a is LowerBound of right_open_halfline a
let x be ext-real number ; :: according to XXREAL_2:def_2 ::_thesis: ( not x in right_open_halfline a or a <= x )
thus  ( not x in right_open_halfline a or a <= x ) by XXREAL_1:235; ::_thesis: verum
end;
thus  not right_open_halfline a is bounded_above by Lm2; ::_thesis: right_open_halfline a is interval
let r, s be ext-real number ; :: according to XXREAL_2:def_12 ::_thesis: ( not r in right_open_halfline a or not s in right_open_halfline a or [.r,s.] c= right_open_halfline a )
assume A11: r in right_open_halfline a ; ::_thesis: ( not s in right_open_halfline a or [.r,s.] c= right_open_halfline a )
then A12: a < r by XXREAL_1:235;
assume  s in right_open_halfline a ; ::_thesis: [.r,s.] c= right_open_halfline a
then reconsider rr = r, ss = s as Real by A11;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.r,s.] or x in right_open_halfline a )
assume A13: x in [.r,s.] ; ::_thesis: x in right_open_halfline a
then  x in [.rr,ss.] ;
then reconsider x = x as Real ;
 r <= x by A13, XXREAL_1:1;
then  a < x by A12, XXREAL_0:2;
hence  x in right_open_halfline a by XXREAL_1:235; ::_thesis: verum
end;
end;

theorem Th9: :: RCOMP_3:9
for a being real number holds upper_bound (left_closed_halfline a) = a
proof
let a be real number ; ::_thesis: upper_bound (left_closed_halfline a) = a
set X = left_closed_halfline a;
A1: for s being real number st 0 < s holds
ex r being real number st
( r in left_closed_halfline a & a - s < r )
proof
let s be real number ; ::_thesis: ( 0 < s implies ex r being real number st
( r in left_closed_halfline a & a - s < r ) )
assume  0 < s ; ::_thesis: ex r being real number st
( r in left_closed_halfline a & a - s < r )
then A2: a - s < a - 0 by XREAL_1:15;
take  a ; ::_thesis: ( a in left_closed_halfline a & a - s < a )
thus  a in left_closed_halfline a by XXREAL_1:234; ::_thesis: a - s < a
thus  a - s < a by A2; ::_thesis: verum
end;
 for r being real number st r in left_closed_halfline a holds
r <= a by XXREAL_1:234;
hence  upper_bound (left_closed_halfline a) = a by A1, SEQ_4:def_1; ::_thesis: verum
end;

theorem Th10: :: RCOMP_3:10
for a being real number holds upper_bound (left_open_halfline a) = a
proof
let a be real number ; ::_thesis: upper_bound (left_open_halfline a) = a
set X = left_open_halfline a;
A1: for s being real number st 0 < s holds
ex r being real number st
( r in left_open_halfline a & a - s < r )
proof
let s be real number ; ::_thesis: ( 0 < s implies ex r being real number st
( r in left_open_halfline a & a - s < r ) )
assume  0 < s ; ::_thesis: ex r being real number st
( r in left_open_halfline a & a - s < r )
then A2: a - s < a - 0 by XREAL_1:15;
take  ((a - s) + a) / 2 ; ::_thesis: ( ((a - s) + a) / 2 in left_open_halfline a & a - s < ((a - s) + a) / 2 )
 ((a - s) + a) / 2 < a by A2, XREAL_1:226;
hence  ( ((a - s) + a) / 2 in left_open_halfline a & a - s < ((a - s) + a) / 2 ) by A2, XREAL_1:226, XXREAL_1:233; ::_thesis: verum
end;
 for r being real number st r in left_open_halfline a holds
r <= a by XXREAL_1:233;
hence  upper_bound (left_open_halfline a) = a by A1, SEQ_4:def_1; ::_thesis: verum
end;

theorem Th11: :: RCOMP_3:11
for a being real number holds lower_bound (right_closed_halfline a) = a
proof
let a be real number ; ::_thesis: lower_bound (right_closed_halfline a) = a
set X = right_closed_halfline a;
A1: for s being real number st 0 < s holds
ex r being real number st
( r in right_closed_halfline a & r < a + s )
proof
let s be real number ; ::_thesis: ( 0 < s implies ex r being real number st
( r in right_closed_halfline a & r < a + s ) )
assume  0 < s ; ::_thesis: ex r being real number st
( r in right_closed_halfline a & r < a + s )
then A2: a + 0 < a + s by XREAL_1:6;
take  a ; ::_thesis: ( a in right_closed_halfline a & a < a + s )
thus  a in right_closed_halfline a by XXREAL_1:236; ::_thesis: a < a + s
thus  a < a + s by A2; ::_thesis: verum
end;
 for r being real number st r in right_closed_halfline a holds
a <= r by XXREAL_1:236;
hence  lower_bound (right_closed_halfline a) = a by A1, SEQ_4:def_2; ::_thesis: verum
end;

theorem Th12: :: RCOMP_3:12
for a being real number holds lower_bound (right_open_halfline a) = a
proof
let a be real number ; ::_thesis: lower_bound (right_open_halfline a) = a
set X = right_open_halfline a;
A1: for s being real number st 0 < s holds
ex r being real number st
( r in right_open_halfline a & r < a + s )
proof
let s be real number ; ::_thesis: ( 0 < s implies ex r being real number st
( r in right_open_halfline a & r < a + s ) )
assume  0 < s ; ::_thesis: ex r being real number st
( r in right_open_halfline a & r < a + s )
then A2: a + 0 < a + s by XREAL_1:6;
take  ((a + a) + s) / 2 ; ::_thesis: ( ((a + a) + s) / 2 in right_open_halfline a & ((a + a) + s) / 2 < a + s )
 a < (a + (a + s)) / 2 by A2, XREAL_1:226;
hence  ( ((a + a) + s) / 2 in right_open_halfline a & ((a + a) + s) / 2 < a + s ) by A2, XREAL_1:226, XXREAL_1:235; ::_thesis: verum
end;
 for r being real number st r in right_open_halfline a holds
a <= r by XXREAL_1:235;
hence  lower_bound (right_open_halfline a) = a by A1, SEQ_4:def_2; ::_thesis: verum
end;

begin

registration
cluster [#] REAL ->  non bounded_below non bounded_above interval ;
coherence
 ( [#] REAL is interval & not [#] REAL is bounded_below & not [#] REAL is bounded_above ) ;
end;

theorem Th13: :: RCOMP_3:13
for X being real-bounded interval Subset of REAL st lower_bound X in X & upper_bound X in X holds
X = [.(lower_bound X),(upper_bound X).]
proof
let X be real-bounded interval Subset of REAL; ::_thesis: ( lower_bound X in X & upper_bound X in X implies X = [.(lower_bound X),(upper_bound X).] )
assume A1: ( lower_bound X in X & upper_bound X in X ) ; ::_thesis: X = [.(lower_bound X),(upper_bound X).]
reconsider X1 = X as non empty real-bounded interval Subset of REAL by A1;
 X1 c= [.(lower_bound X1),(upper_bound X1).] by XXREAL_2:69;
hence  X c= [.(lower_bound X),(upper_bound X).] ; :: according to XBOOLE_0:def_10 ::_thesis: [.(lower_bound X),(upper_bound X).] c= X
thus  [.(lower_bound X),(upper_bound X).] c= X by A1, XXREAL_2:def_12; ::_thesis: verum
end;

theorem Th14: :: RCOMP_3:14
for X being real-bounded Subset of REAL st not lower_bound X in X holds
X c= ].(lower_bound X),(upper_bound X).]
proof
let X be real-bounded Subset of REAL; ::_thesis: ( not lower_bound X in X implies X c= ].(lower_bound X),(upper_bound X).] )
assume A1: not lower_bound X in X ; ::_thesis: X c= ].(lower_bound X),(upper_bound X).]
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in ].(lower_bound X),(upper_bound X).] )
assume A2: x in X ; ::_thesis: x in ].(lower_bound X),(upper_bound X).]
then reconsider x = x as Real ;
 lower_bound X <= x by A2, SEQ_4:def_2;
then A3: lower_bound X < x by A1, A2, XXREAL_0:1;
 x <= upper_bound X by A2, SEQ_4:def_1;
hence  x in ].(lower_bound X),(upper_bound X).] by A3, XXREAL_1:2; ::_thesis: verum
end;

theorem Th15: :: RCOMP_3:15
for X being real-bounded interval Subset of REAL st not lower_bound X in X & upper_bound X in X holds
X = ].(lower_bound X),(upper_bound X).]
proof
let X be real-bounded interval Subset of REAL; ::_thesis: ( not lower_bound X in X & upper_bound X in X implies X = ].(lower_bound X),(upper_bound X).] )
assume that
A1: not lower_bound X in X and
A2: upper_bound X in X ; ::_thesis: X = ].(lower_bound X),(upper_bound X).]
thus  X c= ].(lower_bound X),(upper_bound X).] by A1, Th14; :: according to XBOOLE_0:def_10 ::_thesis: ].(lower_bound X),(upper_bound X).] c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].(lower_bound X),(upper_bound X).] or x in X )
assume A3: x in ].(lower_bound X),(upper_bound X).] ; ::_thesis: x in X
then reconsider x = x as Real ;
 lower_bound X < x by A3, XXREAL_1:2;
then  (lower_bound X) - (lower_bound X) < x - (lower_bound X) by XREAL_1:14;
then consider r being real number  such that
A4: r in X and
A5: r < (lower_bound X) + (x - (lower_bound X)) by A2, SEQ_4:def_2;
 x <= upper_bound X by A3, XXREAL_1:2;
then A6: x in [.r,(upper_bound X).] by A5, XXREAL_1:1;
 [.r,(upper_bound X).] c= X by A2, A4, XXREAL_2:def_12;
hence  x in X by A6; ::_thesis: verum
end;

theorem Th16: :: RCOMP_3:16
for X being real-bounded Subset of REAL st not upper_bound X in X holds
X c= [.(lower_bound X),(upper_bound X).[
proof
let X be real-bounded Subset of REAL; ::_thesis: ( not upper_bound X in X implies X c= [.(lower_bound X),(upper_bound X).[ )
assume A1: not upper_bound X in X ; ::_thesis: X c= [.(lower_bound X),(upper_bound X).[
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in [.(lower_bound X),(upper_bound X).[ )
assume A2: x in X ; ::_thesis: x in [.(lower_bound X),(upper_bound X).[
then reconsider x = x as Real ;
 x <= upper_bound X by A2, SEQ_4:def_1;
then A3: x < upper_bound X by A1, A2, XXREAL_0:1;
 lower_bound X <= x by A2, SEQ_4:def_2;
hence  x in [.(lower_bound X),(upper_bound X).[ by A3, XXREAL_1:3; ::_thesis: verum
end;

theorem Th17: :: RCOMP_3:17
for X being real-bounded interval Subset of REAL st lower_bound X in X & not upper_bound X in X holds
X = [.(lower_bound X),(upper_bound X).[
proof
let X be real-bounded interval Subset of REAL; ::_thesis: ( lower_bound X in X & not upper_bound X in X implies X = [.(lower_bound X),(upper_bound X).[ )
assume that
A1: lower_bound X in X and
A2: not upper_bound X in X ; ::_thesis: X = [.(lower_bound X),(upper_bound X).[
thus  X c= [.(lower_bound X),(upper_bound X).[ by A2, Th16; :: according to XBOOLE_0:def_10 ::_thesis: [.(lower_bound X),(upper_bound X).[ c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [.(lower_bound X),(upper_bound X).[ or x in X )
assume A3: x in [.(lower_bound X),(upper_bound X).[ ; ::_thesis: x in X
then reconsider x = x as Real ;
 x < upper_bound X by A3, XXREAL_1:3;
then  x - x < (upper_bound X) - x by XREAL_1:14;
then consider r being real number  such that
A4: r in X and
A5: (upper_bound X) - ((upper_bound X) - x) < r by A1, SEQ_4:def_1;
 lower_bound X <= x by A3, XXREAL_1:3;
then A6: x in [.(lower_bound X),r.] by A5, XXREAL_1:1;
 [.(lower_bound X),r.] c= X by A1, A4, XXREAL_2:def_12;
hence  x in X by A6; ::_thesis: verum
end;

theorem Th18: :: RCOMP_3:18
for X being real-bounded Subset of REAL st not lower_bound X in X & not upper_bound X in X holds
X c= ].(lower_bound X),(upper_bound X).[
proof
let X be real-bounded Subset of REAL; ::_thesis: ( not lower_bound X in X & not upper_bound X in X implies X c= ].(lower_bound X),(upper_bound X).[ )
assume that
A1: not lower_bound X in X and
A2: not upper_bound X in X ; ::_thesis: X c= ].(lower_bound X),(upper_bound X).[
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in ].(lower_bound X),(upper_bound X).[ )
assume A3: x in X ; ::_thesis: x in ].(lower_bound X),(upper_bound X).[
then reconsider x = x as Real ;
 x <= upper_bound X by A3, SEQ_4:def_1;
then A4: x < upper_bound X by A2, A3, XXREAL_0:1;
 lower_bound X <= x by A3, SEQ_4:def_2;
then  lower_bound X < x by A1, A3, XXREAL_0:1;
hence  x in ].(lower_bound X),(upper_bound X).[ by A4, XXREAL_1:4; ::_thesis: verum
end;

theorem Th19: :: RCOMP_3:19
for X being non empty real-bounded interval Subset of REAL st not lower_bound X in X & not upper_bound X in X holds
X = ].(lower_bound X),(upper_bound X).[
proof
let X be non empty real-bounded interval Subset of REAL; ::_thesis: ( not lower_bound X in X & not upper_bound X in X implies X = ].(lower_bound X),(upper_bound X).[ )
assume  ( not lower_bound X in X & not upper_bound X in X ) ; ::_thesis: X = ].(lower_bound X),(upper_bound X).[
hence  X c= ].(lower_bound X),(upper_bound X).[ by Th18; :: according to XBOOLE_0:def_10 ::_thesis: ].(lower_bound X),(upper_bound X).[ c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ].(lower_bound X),(upper_bound X).[ or x in X )
assume A1: x in ].(lower_bound X),(upper_bound X).[ ; ::_thesis: x in X
then reconsider x = x as Real ;
 lower_bound X < x by A1, XXREAL_1:4;
then  (lower_bound X) - (lower_bound X) < x - (lower_bound X) by XREAL_1:14;
then consider s being real number  such that
A2: ( s in X & s < (lower_bound X) + (x - (lower_bound X)) ) by SEQ_4:def_2;
 x < upper_bound X by A1, XXREAL_1:4;
then  x - x < (upper_bound X) - x by XREAL_1:14;
then consider r being real number  such that
A3: ( r in X & (upper_bound X) - ((upper_bound X) - x) < r ) by SEQ_4:def_1;
 ( [.s,r.] c= X & x in [.s,r.] ) by A2, A3, XXREAL_1:1, XXREAL_2:def_12;
hence  x in X ; ::_thesis: verum
end;

theorem Th20: :: RCOMP_3:20
for X being Subset of REAL st X is bounded_above holds
X c= left_closed_halfline (upper_bound X)
proof
let X be Subset of REAL; ::_thesis: ( X is bounded_above implies X c= left_closed_halfline (upper_bound X) )
assume A1: X is bounded_above ; ::_thesis: X c= left_closed_halfline (upper_bound X)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in left_closed_halfline (upper_bound X) )
assume A2: x in X ; ::_thesis: x in left_closed_halfline (upper_bound X)
then reconsider x = x as Real ;
 x <= upper_bound X by A1, A2, SEQ_4:def_1;
hence  x in left_closed_halfline (upper_bound X) by XXREAL_1:234; ::_thesis: verum
end;

theorem Th21: :: RCOMP_3:21
for X being interval Subset of REAL st not X is bounded_below & X is bounded_above & upper_bound X in X holds
X = left_closed_halfline (upper_bound X)
proof
let X be interval Subset of REAL; ::_thesis: ( not X is bounded_below & X is bounded_above & upper_bound X in X implies X = left_closed_halfline (upper_bound X) )
assume that
A1: not X is bounded_below and
A2: X is bounded_above and
A3: upper_bound X in X ; ::_thesis: X = left_closed_halfline (upper_bound X)
thus  X c= left_closed_halfline (upper_bound X) by A2, Th20; :: according to XBOOLE_0:def_10 ::_thesis: left_closed_halfline (upper_bound X) c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in left_closed_halfline (upper_bound X) or x in X )
assume A4: x in left_closed_halfline (upper_bound X) ; ::_thesis: x in X
then reconsider x = x as Real ;
 x is not LowerBound of X by A1, XXREAL_2:def_9;
then consider r being ext-real number  such that
A5: r in X and
A6: x > r by XXREAL_2:def_2;
reconsider r = r as real number by A5;
 x <= upper_bound X by A4, XXREAL_1:234;
then A7: x in [.r,(upper_bound X).] by A6, XXREAL_1:1;
 [.r,(upper_bound X).] c= X by A3, A5, XXREAL_2:def_12;
hence  x in X by A7; ::_thesis: verum
end;

theorem Th22: :: RCOMP_3:22
for X being Subset of REAL st X is bounded_above & not upper_bound X in X holds
X c= left_open_halfline (upper_bound X)
proof
let X be Subset of REAL; ::_thesis: ( X is bounded_above & not upper_bound X in X implies X c= left_open_halfline (upper_bound X) )
assume that
A1: X is bounded_above and
A2: not upper_bound X in X ; ::_thesis: X c= left_open_halfline (upper_bound X)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in left_open_halfline (upper_bound X) )
assume A3: x in X ; ::_thesis: x in left_open_halfline (upper_bound X)
then reconsider x = x as Real ;
 x <= upper_bound X by A1, A3, SEQ_4:def_1;
then  x < upper_bound X by A2, A3, XXREAL_0:1;
hence  x in left_open_halfline (upper_bound X) by XXREAL_1:233; ::_thesis: verum
end;

theorem Th23: :: RCOMP_3:23
for X being non empty interval Subset of REAL st not X is bounded_below & X is bounded_above & not upper_bound X in X holds
X = left_open_halfline (upper_bound X)
proof
let X be non empty interval Subset of REAL; ::_thesis: ( not X is bounded_below & X is bounded_above & not upper_bound X in X implies X = left_open_halfline (upper_bound X) )
assume that
A1: not X is bounded_below and
A2: X is bounded_above and
A3: not upper_bound X in X ; ::_thesis: X = left_open_halfline (upper_bound X)
thus  X c= left_open_halfline (upper_bound X) by A2, A3, Th22; :: according to XBOOLE_0:def_10 ::_thesis: left_open_halfline (upper_bound X) c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in left_open_halfline (upper_bound X) or x in X )
assume A4: x in left_open_halfline (upper_bound X) ; ::_thesis: x in X
then reconsider x = x as Real ;
 x is not LowerBound of X by A1, XXREAL_2:def_9;
then consider r being ext-real number  such that
A5: ( r in X & x > r ) by XXREAL_2:def_2;
reconsider r = r as real number by A5;
 x < upper_bound X by A4, XXREAL_1:233;
then  x - x < (upper_bound X) - x by XREAL_1:14;
then consider s being real number  such that
A6: ( s in X & (upper_bound X) - ((upper_bound X) - x) < s ) by A2, SEQ_4:def_1;
 ( [.r,s.] c= X & x in [.r,s.] ) by A5, A6, XXREAL_1:1, XXREAL_2:def_12;
hence  x in X ; ::_thesis: verum
end;

theorem Th24: :: RCOMP_3:24
for X being Subset of REAL st X is bounded_below holds
X c= right_closed_halfline (lower_bound X)
proof
let X be Subset of REAL; ::_thesis: ( X is bounded_below implies X c= right_closed_halfline (lower_bound X) )
assume A1: X is bounded_below ; ::_thesis: X c= right_closed_halfline (lower_bound X)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in right_closed_halfline (lower_bound X) )
assume A2: x in X ; ::_thesis: x in right_closed_halfline (lower_bound X)
then reconsider x = x as Real ;
 lower_bound X <= x by A1, A2, SEQ_4:def_2;
hence  x in right_closed_halfline (lower_bound X) by XXREAL_1:236; ::_thesis: verum
end;

theorem Th25: :: RCOMP_3:25
for X being interval Subset of REAL st X is bounded_below & not X is bounded_above & lower_bound X in X holds
X = right_closed_halfline (lower_bound X)
proof
let X be interval Subset of REAL; ::_thesis: ( X is bounded_below & not X is bounded_above & lower_bound X in X implies X = right_closed_halfline (lower_bound X) )
assume that
A1: X is bounded_below and
A2: not X is bounded_above and
A3: lower_bound X in X ; ::_thesis: X = right_closed_halfline (lower_bound X)
thus  X c= right_closed_halfline (lower_bound X) by A1, Th24; :: according to XBOOLE_0:def_10 ::_thesis: right_closed_halfline (lower_bound X) c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in right_closed_halfline (lower_bound X) or x in X )
assume A4: x in right_closed_halfline (lower_bound X) ; ::_thesis: x in X
then reconsider x = x as Real ;
 x is not UpperBound of X by A2, XXREAL_2:def_10;
then consider r being ext-real number  such that
A5: r in X and
A6: x < r by XXREAL_2:def_1;
reconsider r = r as real number by A5;
 lower_bound X <= x by A4, XXREAL_1:236;
then A7: x in [.(lower_bound X),r.] by A6, XXREAL_1:1;
 [.(lower_bound X),r.] c= X by A3, A5, XXREAL_2:def_12;
hence  x in X by A7; ::_thesis: verum
end;

theorem Th26: :: RCOMP_3:26
for X being Subset of REAL st X is bounded_below & not lower_bound X in X holds
X c= right_open_halfline (lower_bound X)
proof
let X be Subset of REAL; ::_thesis: ( X is bounded_below & not lower_bound X in X implies X c= right_open_halfline (lower_bound X) )
assume that
A1: X is bounded_below and
A2: not lower_bound X in X ; ::_thesis: X c= right_open_halfline (lower_bound X)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in right_open_halfline (lower_bound X) )
assume A3: x in X ; ::_thesis: x in right_open_halfline (lower_bound X)
then reconsider x = x as Real ;
 lower_bound X <= x by A1, A3, SEQ_4:def_2;
then  lower_bound X < x by A2, A3, XXREAL_0:1;
hence  x in right_open_halfline (lower_bound X) by XXREAL_1:235; ::_thesis: verum
end;

theorem Th27: :: RCOMP_3:27
for X being non empty interval Subset of REAL st X is bounded_below & not X is bounded_above & not lower_bound X in X holds
X = right_open_halfline (lower_bound X)
proof
let X be non empty interval Subset of REAL; ::_thesis: ( X is bounded_below & not X is bounded_above & not lower_bound X in X implies X = right_open_halfline (lower_bound X) )
assume that
A1: X is bounded_below and
A2: not X is bounded_above and
A3: not lower_bound X in X ; ::_thesis: X = right_open_halfline (lower_bound X)
thus  X c= right_open_halfline (lower_bound X) by A1, A3, Th26; :: according to XBOOLE_0:def_10 ::_thesis: right_open_halfline (lower_bound X) c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in right_open_halfline (lower_bound X) or x in X )
assume A4: x in right_open_halfline (lower_bound X) ; ::_thesis: x in X
then reconsider x = x as Real ;
 x is not UpperBound of X by A2, XXREAL_2:def_10;
then consider r being ext-real number  such that
A5: ( r in X & x < r ) by XXREAL_2:def_1;
 lower_bound X < x by A4, XXREAL_1:235;
then  (lower_bound X) - (lower_bound X) < x - (lower_bound X) by XREAL_1:14;
then consider s being real number  such that
A6: ( s in X & s < (lower_bound X) + (x - (lower_bound X)) ) by A1, SEQ_4:def_2;
reconsider r = r as real number by A5;
 ( [.s,r.] c= X & x in [.s,r.] ) by A5, A6, XXREAL_1:1, XXREAL_2:def_12;
hence  x in X ; ::_thesis: verum
end;

theorem Th28: :: RCOMP_3:28
for X being interval Subset of REAL st not X is bounded_above & not X is bounded_below holds
X = REAL
proof
let X be interval Subset of REAL; ::_thesis: ( not X is bounded_above & not X is bounded_below implies X = REAL )
assume that
A1: not X is bounded_above and
A2: not X is bounded_below ; ::_thesis: X = REAL
thus  X c= REAL ; :: according to XBOOLE_0:def_10 ::_thesis: REAL c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in REAL or x in X )
assume  x in REAL ; ::_thesis: x in X
then reconsider x = x as Real ;
 x is not UpperBound of X by A1, XXREAL_2:def_10;
then consider r being ext-real number  such that
A3: ( r in X & r > x ) by XXREAL_2:def_1;
reconsider r = r as real number by A3;
 x is not LowerBound of X by A2, XXREAL_2:def_9;
then consider s being ext-real number  such that
A4: ( s in X & s < x ) by XXREAL_2:def_2;
reconsider s = s as real number by A4;
 ( [.s,r.] c= X & x in [.s,r.] ) by A3, A4, XXREAL_1:1, XXREAL_2:def_12;
hence  x in X ; ::_thesis: verum
end;

theorem Th29: :: RCOMP_3:29
for X being interval Subset of REAL holds
( X is empty or X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
proof
let X be interval Subset of REAL; ::_thesis: ( X is empty or X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
assume  not X is empty ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then reconsider X = X as non empty interval Subset of REAL ;
percases ( X is real-bounded or not X is real-bounded ) ;
suppose X is real-bounded ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then reconsider X = X as non empty real-bounded interval Subset of REAL ;
percases ( X is trivial or not X is trivial ) ;
suppose X is trivial ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then consider x being set  such that
A1: X = {x} by ZFMISC_1:131;
 x in X by A1, TARSKI:def_1;
then reconsider x = x as Real ;
 X = [.x,x.] by A1, XXREAL_1:17;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
suppose not X is trivial ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  ex p, q being set st
( p in X & q in X & p <> q ) by ZFMISC_1:def_10;
then A2: lower_bound X < upper_bound X by SEQ_4:12;
percases ( ( upper_bound X in X & lower_bound X in X ) or ( upper_bound X in X & not lower_bound X in X ) or ( not upper_bound X in X & lower_bound X in X ) or ( not upper_bound X in X & not lower_bound X in X ) ) ;
suppose ( upper_bound X in X & lower_bound X in X ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = [.(lower_bound X),(upper_bound X).] by Th13;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by A2; ::_thesis: verum
end;
suppose ( upper_bound X in X & not lower_bound X in X ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = ].(lower_bound X),(upper_bound X).] by Th15;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by A2; ::_thesis: verum
end;
suppose ( not upper_bound X in X & lower_bound X in X ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = [.(lower_bound X),(upper_bound X).[ by Th17;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by A2; ::_thesis: verum
end;
suppose ( not upper_bound X in X & not lower_bound X in X ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = ].(lower_bound X),(upper_bound X).[ by Th19;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by A2; ::_thesis: verum
end;
end;
end;
end;
end;
supposeA3: not X is real-bounded ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
percases ( ( not X is bounded_below & X is bounded_above ) or ( not X is bounded_above & X is bounded_below ) or ( not X is bounded_above & not X is bounded_below ) ) by A3;
supposeA4: ( not X is bounded_below & X is bounded_above ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
percases ( upper_bound X in X or not upper_bound X in X ) ;
suppose upper_bound X in X ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = left_closed_halfline (upper_bound X) by A4, Th21;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
suppose not upper_bound X in X ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = left_open_halfline (upper_bound X) by A4, Th23;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
end;
end;
supposeA5: ( not X is bounded_above & X is bounded_below ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
percases ( lower_bound X in X or not lower_bound X in X ) ;
suppose lower_bound X in X ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = right_closed_halfline (lower_bound X) by A5, Th25;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
suppose not lower_bound X in X ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
then  X = right_open_halfline (lower_bound X) by A5, Th27;
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
end;
end;
suppose ( not X is bounded_above & not X is bounded_below ) ; ::_thesis: ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) )
hence  ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by Th28; ::_thesis: verum
end;
end;
end;
end;
end;

theorem Th30: :: RCOMP_3:30
for r being real number
for X being non empty interval Subset of REAL holds
( r in X or r <= lower_bound X or upper_bound X <= r )
proof
let r be real number ; ::_thesis: for X being non empty interval Subset of REAL holds
( r in X or r <= lower_bound X or upper_bound X <= r )
let X be non empty interval Subset of REAL; ::_thesis: ( r in X or r <= lower_bound X or upper_bound X <= r )
assume A1: not r in X ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
percases ( X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by Th29;
suppose X = REAL ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, XREAL_0:def_1; ::_thesis: verum
end;
suppose ex a being real number st X = left_closed_halfline a ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a being real number  such that
A2: X = left_closed_halfline a ;
 upper_bound X = a by A2, Th9;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A2, XXREAL_1:234; ::_thesis: verum
end;
suppose ex a being real number st X = left_open_halfline a ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a being real number  such that
A3: X = left_open_halfline a ;
 upper_bound X = a by A3, Th10;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A3, XXREAL_1:233; ::_thesis: verum
end;
suppose ex a being real number st X = right_closed_halfline a ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a being real number  such that
A4: X = right_closed_halfline a ;
 lower_bound X = a by A4, Th11;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A4, XXREAL_1:236; ::_thesis: verum
end;
suppose ex a being real number st X = right_open_halfline a ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a being real number  such that
A5: X = right_open_halfline a ;
 lower_bound X = a by A5, Th12;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A5, XXREAL_1:235; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & X = [.a,b.] ) ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a, b being real number  such that
A6: a <= b and
A7: X = [.a,b.] ;
 ( lower_bound X = a & upper_bound X = b ) by A6, A7, JORDAN5A:19;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A7, XXREAL_1:1; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = [.a,b.[ ) ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a, b being real number  such that
A8: a < b and
A9: X = [.a,b.[ ;
 ( lower_bound X = a & upper_bound X = b ) by A8, A9, Th4, Th5;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A9, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.] ) ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a, b being real number  such that
A10: a < b and
A11: X = ].a,b.] ;
 ( lower_bound X = a & upper_bound X = b ) by A10, A11, Th6, Th7;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A11, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.[ ) ; ::_thesis: ( r <= lower_bound X or upper_bound X <= r )
then consider a, b being real number  such that
A12: a < b and
A13: X = ].a,b.[ ;
 ( lower_bound X = a & upper_bound X = b ) by A12, A13, TOPREAL6:17;
hence  ( r <= lower_bound X or upper_bound X <= r ) by A1, A13, XXREAL_1:4; ::_thesis: verum
end;
end;
end;

theorem Th31: :: RCOMP_3:31
for X, Y being non empty real-bounded interval Subset of REAL st lower_bound X <= lower_bound Y & upper_bound Y <= upper_bound X & ( lower_bound X = lower_bound Y & lower_bound Y in Y implies lower_bound X in X ) & ( upper_bound X = upper_bound Y & upper_bound Y in Y implies upper_bound X in X ) holds
Y c= X
proof
let X, Y be non empty real-bounded interval Subset of REAL; ::_thesis: ( lower_bound X <= lower_bound Y & upper_bound Y <= upper_bound X & ( lower_bound X = lower_bound Y & lower_bound Y in Y implies lower_bound X in X ) & ( upper_bound X = upper_bound Y & upper_bound Y in Y implies upper_bound X in X ) implies Y c= X )
assume that
A1: lower_bound X <= lower_bound Y and
A2: upper_bound Y <= upper_bound X and
A3: ( lower_bound X = lower_bound Y & lower_bound Y in Y implies lower_bound X in X ) and
A4: ( upper_bound X = upper_bound Y & upper_bound Y in Y implies upper_bound X in X ) ; ::_thesis: Y c= X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Y or x in X )
assume A5: x in Y ; ::_thesis: x in X
then reconsider x = x as Real ;
A6: Y c= [.(lower_bound Y),(upper_bound Y).] by XXREAL_2:69;
then A7: lower_bound Y <= x by A5, XXREAL_1:1;
then A8: lower_bound X <= x by A1, XXREAL_0:2;
A9: x <= upper_bound Y by A5, A6, XXREAL_1:1;
then A10: x <= upper_bound X by A2, XXREAL_0:2;
percases ( X = [#] REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by Th29;
suppose X = [#] REAL ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ( ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a ) ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & X = [.a,b.] ) ; ::_thesis: x in X
then consider a, b being real number  such that
A11: a <= b and
A12: X = [.a,b.] ;
 ( lower_bound X = a & upper_bound X = b ) by A11, A12, JORDAN5A:19;
hence  x in X by A8, A10, A12, XXREAL_1:1; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = [.a,b.[ ) ; ::_thesis: x in X
then consider a, b being real number  such that
A13: a < b and
A14: X = [.a,b.[ ;
A15: lower_bound X = a by A13, A14, Th4;
A16: upper_bound X = b by A13, A14, Th5;
percases ( Y = [#] REAL or ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a or ex a, b being real number st
( a <= b & Y = [.a,b.] ) or ex a, b being real number st
( a < b & Y = [.a,b.[ ) or ex a, b being real number st
( a < b & Y = ].a,b.] ) or ex a, b being real number st
( a < b & Y = ].a,b.[ ) ) by Th29;
suppose Y = [#] REAL ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ( ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a ) ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & Y = [.a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A17: ( a1 <= b1 & Y = [.a1,b1.] ) ;
A18: upper_bound Y = b1 by A17, JORDAN5A:19;
then  b1 < b by A2, A4, A14, A16, A17, XXREAL_0:1, XXREAL_1:1, XXREAL_1:3;
then  x < b by A9, A18, XXREAL_0:2;
hence  x in X by A8, A14, A15, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = [.a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A19: ( a1 < b1 & Y = [.a1,b1.[ ) ;
 ( upper_bound Y = b1 & x < b1 ) by A5, A19, Th5, XXREAL_1:3;
then  x < b by A2, A16, XXREAL_0:2;
hence  x in X by A8, A14, A15, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A20: ( a1 < b1 & Y = ].a1,b1.] ) ;
A21: upper_bound Y = b1 by A20, Th7;
then  b1 < b by A2, A4, A14, A16, A20, XXREAL_0:1, XXREAL_1:2, XXREAL_1:3;
then  x < b by A9, A21, XXREAL_0:2;
hence  x in X by A8, A14, A15, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A22: ( a1 < b1 & Y = ].a1,b1.[ ) ;
 ( upper_bound Y = b1 & x < b1 ) by A5, A22, TOPREAL6:17, XXREAL_1:4;
then  x < b by A2, A16, XXREAL_0:2;
hence  x in X by A8, A14, A15, XXREAL_1:3; ::_thesis: verum
end;
end;
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.] ) ; ::_thesis: x in X
then consider a, b being real number  such that
A23: a < b and
A24: X = ].a,b.] ;
A25: lower_bound X = a by A23, A24, Th6;
A26: upper_bound X = b by A23, A24, Th7;
percases ( Y = [#] REAL or ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a or ex a, b being real number st
( a <= b & Y = [.a,b.] ) or ex a, b being real number st
( a < b & Y = [.a,b.[ ) or ex a, b being real number st
( a < b & Y = ].a,b.] ) or ex a, b being real number st
( a < b & Y = ].a,b.[ ) ) by Th29;
suppose Y = [#] REAL ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ( ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a ) ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & Y = [.a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A27: ( a1 <= b1 & Y = [.a1,b1.] ) ;
A28: lower_bound Y = a1 by A27, JORDAN5A:19;
then  a < a1 by A1, A3, A24, A25, A27, XXREAL_0:1, XXREAL_1:1, XXREAL_1:2;
then  a < x by A7, A28, XXREAL_0:2;
hence  x in X by A10, A24, A26, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = [.a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A29: a1 < b1 and
A30: Y = [.a1,b1.[ ;
 lower_bound Y = a1 by A29, A30, Th4;
then A31: a < a1 by A1, A3, A24, A25, A29, A30, XXREAL_0:1, XXREAL_1:2, XXREAL_1:3;
 a1 <= x by A5, A30, XXREAL_1:3;
then  a < x by A31, XXREAL_0:2;
hence  x in X by A10, A24, A26, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A32: ( a1 < b1 & Y = ].a1,b1.] ) ;
 ( lower_bound Y = a1 & a1 < x ) by A5, A32, Th6, XXREAL_1:2;
then  a < x by A1, A25, XXREAL_0:2;
hence  x in X by A10, A24, A26, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A33: ( a1 < b1 & Y = ].a1,b1.[ ) ;
 ( lower_bound Y = a1 & a1 < x ) by A5, A33, TOPREAL6:17, XXREAL_1:4;
then  a < x by A1, A25, XXREAL_0:2;
hence  x in X by A10, A24, A26, XXREAL_1:2; ::_thesis: verum
end;
end;
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.[ ) ; ::_thesis: x in X
then consider a, b being real number  such that
A34: a < b and
A35: X = ].a,b.[ ;
A36: lower_bound X = a by A34, A35, TOPREAL6:17;
A37: upper_bound X = b by A34, A35, TOPREAL6:17;
percases ( Y = [#] REAL or ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a or ex a, b being real number st
( a <= b & Y = [.a,b.] ) or ex a, b being real number st
( a < b & Y = [.a,b.[ ) or ex a, b being real number st
( a < b & Y = ].a,b.] ) or ex a, b being real number st
( a < b & Y = ].a,b.[ ) ) by Th29;
suppose Y = [#] REAL ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ( ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a ) ; ::_thesis: x in X
hence  x in X ; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & Y = [.a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A38: a1 <= b1 and
A39: Y = [.a1,b1.] ;
 upper_bound Y = b1 by A38, A39, JORDAN5A:19;
then A40: b1 < b by A2, A4, A35, A37, A38, A39, XXREAL_0:1, XXREAL_1:1, XXREAL_1:4;
 x <= b1 by A5, A39, XXREAL_1:1;
then A41: x < b by A40, XXREAL_0:2;
A42: lower_bound Y = a1 by A38, A39, JORDAN5A:19;
then  a < a1 by A1, A3, A35, A36, A38, A39, XXREAL_0:1, XXREAL_1:1, XXREAL_1:4;
then  a < x by A7, A42, XXREAL_0:2;
hence  x in X by A35, A41, XXREAL_1:4; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = [.a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A43: a1 < b1 and
A44: Y = [.a1,b1.[ ;
 lower_bound Y = a1 by A43, A44, Th4;
then A45: a < a1 by A1, A3, A35, A36, A43, A44, XXREAL_0:1, XXREAL_1:3, XXREAL_1:4;
 a1 <= x by A5, A44, XXREAL_1:3;
then A46: a < x by A45, XXREAL_0:2;
 ( upper_bound Y = b1 & x < b1 ) by A5, A43, A44, Th5, XXREAL_1:3;
then  x < b by A2, A37, XXREAL_0:2;
hence  x in X by A35, A46, XXREAL_1:4; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.] ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A47: a1 < b1 and
A48: Y = ].a1,b1.] ;
 upper_bound Y = b1 by A47, A48, Th7;
then A49: b1 < b by A2, A4, A35, A37, A47, A48, XXREAL_0:1, XXREAL_1:2, XXREAL_1:4;
 x <= b1 by A5, A48, XXREAL_1:2;
then A50: x < b by A49, XXREAL_0:2;
 ( lower_bound Y = a1 & a1 < x ) by A5, A47, A48, Th6, XXREAL_1:2;
then  a < x by A1, A36, XXREAL_0:2;
hence  x in X by A35, A50, XXREAL_1:4; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.[ ) ; ::_thesis: x in X
then consider a1, b1 being real number  such that
A51: ( a1 < b1 & Y = ].a1,b1.[ ) ;
 ( lower_bound Y = a1 & a1 < x ) by A5, A51, TOPREAL6:17, XXREAL_1:4;
then A52: a < x by A1, A36, XXREAL_0:2;
 ( upper_bound Y = b1 & x < b1 ) by A5, A51, TOPREAL6:17, XXREAL_1:4;
then  x < b by A2, A37, XXREAL_0:2;
hence  x in X by A35, A52, XXREAL_1:4; ::_thesis: verum
end;
end;
end;
end;
end;

registration
cluster non empty complex-membered ext-real-membered real-membered closed open non real-bounded interval for Element of K32(REAL);
existence
 ex b1 being Subset of REAL st
( b1 is open & b1 is closed & b1 is interval & not b1 is empty & not b1 is real-bounded )
proof
take  [#] REAL ; ::_thesis: ( [#] REAL is open & [#] REAL is closed & [#] REAL is interval & not [#] REAL is empty & not [#] REAL is real-bounded )
thus  ( [#] REAL is open & [#] REAL is closed & [#] REAL is interval & not [#] REAL is empty & not [#] REAL is real-bounded ) ; ::_thesis: verum
end;
end;

begin

theorem Th32: :: RCOMP_3:32
for a, b being real number
for X being Subset of R^1 st a <= b & X = [.a,b.] holds
Fr X = {a,b}
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st a <= b & X = [.a,b.] holds
Fr X = {a,b}
let X be Subset of R^1; ::_thesis: ( a <= b & X = [.a,b.] implies Fr X = {a,b} )
assume that
A1: a <= b and
A2: X = [.a,b.] ; ::_thesis: Fr X = {a,b}
A3: Cl X = Cl [.a,b.] by A2, JORDAN5A:24
.= [.a,b.] by MEASURE6:59 ;
A4: [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} by A1, Th8;
set LO = R^1 (left_open_halfline a);
set RC = R^1 (right_closed_halfline b);
set RO = R^1 (right_open_halfline b);
set LC = R^1 (left_closed_halfline a);
A5: R^1 (right_closed_halfline b) = right_closed_halfline b by TOPREALB:def_3;
A6: R^1 (left_closed_halfline a) = left_closed_halfline a by TOPREALB:def_3;
A7: R^1 (right_open_halfline b) = right_open_halfline b by TOPREALB:def_3;
A8: R^1 (left_open_halfline a) = left_open_halfline a by TOPREALB:def_3;
then A9: [.a,b.] ` = (R^1 (left_open_halfline a)) \/ (R^1 (right_open_halfline b)) by A7, XXREAL_1:385;
Cl (X `) = Cl ([.a,b.] `) by A2, JORDAN5A:24, TOPMETR:17
.= (Cl (left_open_halfline a)) \/ (Cl (right_open_halfline b)) by A8, A7, A9, Th3
.= (Cl (R^1 (left_open_halfline a))) \/ (Cl (right_open_halfline b)) by A8, JORDAN5A:24
.= (Cl (R^1 (left_open_halfline a))) \/ (Cl (R^1 (right_open_halfline b))) by A7, JORDAN5A:24
.= (R^1 (left_closed_halfline a)) \/ (Cl (R^1 (right_open_halfline b))) by A6, BORSUK_5:51, TOPREALB:def_3
.= (R^1 (left_closed_halfline a)) \/ (R^1 (right_closed_halfline b)) by A5, BORSUK_5:49, TOPREALB:def_3
.= (left_closed_halfline a) \/ (right_closed_halfline b) by A5, TOPREALB:def_3 ;
hence  Fr X = {a,b} by A3, A4, TOPS_1:def_2; ::_thesis: verum
end;

theorem :: RCOMP_3:33
for a, b being real number
for X being Subset of R^1 st a < b & X = ].a,b.[ holds
Fr X = {a,b}
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st a < b & X = ].a,b.[ holds
Fr X = {a,b}
let X be Subset of R^1; ::_thesis: ( a < b & X = ].a,b.[ implies Fr X = {a,b} )
assume that
A1: a < b and
A2: X = ].a,b.[ ; ::_thesis: Fr X = {a,b}
A3: Cl X = Cl ].a,b.[ by A2, JORDAN5A:24
.= [.a,b.] by A1, JORDAN5A:26 ;
set RC = R^1 (right_closed_halfline b);
set LC = R^1 (left_closed_halfline a);
A4: ( R^1 (right_closed_halfline b) = right_closed_halfline b & R^1 (left_closed_halfline a) = left_closed_halfline a ) by TOPREALB:def_3;
then A5: ].a,b.[ ` = (R^1 (left_closed_halfline a)) \/ (R^1 (right_closed_halfline b)) by XXREAL_1:398;
A6: [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} by A1, Th8;
Cl (X `) = Cl (].a,b.[ `) by A2, JORDAN5A:24, TOPMETR:17
.= (Cl (left_closed_halfline a)) \/ (Cl (right_closed_halfline b)) by A4, A5, Th3
.= (Cl (left_closed_halfline a)) \/ (right_closed_halfline b) by MEASURE6:59
.= (left_closed_halfline a) \/ (right_closed_halfline b) by MEASURE6:59 ;
hence  Fr X = {a,b} by A3, A6, TOPS_1:def_2; ::_thesis: verum
end;

theorem Th34: :: RCOMP_3:34
for a, b being real number
for X being Subset of R^1 st a < b & X = [.a,b.[ holds
Fr X = {a,b}
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st a < b & X = [.a,b.[ holds
Fr X = {a,b}
let X be Subset of R^1; ::_thesis: ( a < b & X = [.a,b.[ implies Fr X = {a,b} )
assume that
A1: a < b and
A2: X = [.a,b.[ ; ::_thesis: Fr X = {a,b}
A3: ( Cl X = [.a,b.] & [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} ) by A1, A2, Th8, BORSUK_5:35;
set LO = R^1 (left_open_halfline a);
set RC = R^1 (right_closed_halfline b);
set LC = R^1 (left_closed_halfline a);
A4: R^1 (right_closed_halfline b) = right_closed_halfline b by TOPREALB:def_3;
A5: R^1 (left_open_halfline a) = left_open_halfline a by TOPREALB:def_3;
then A6: [.a,b.[ ` = (R^1 (left_open_halfline a)) \/ (R^1 (right_closed_halfline b)) by A4, XXREAL_1:382;
A7: R^1 (left_closed_halfline a) = left_closed_halfline a by TOPREALB:def_3;
Cl (X `) = Cl ([.a,b.[ `) by A2, JORDAN5A:24, TOPMETR:17
.= (Cl (left_open_halfline a)) \/ (Cl (right_closed_halfline b)) by A5, A4, A6, Th3
.= (Cl (R^1 (left_open_halfline a))) \/ (Cl (right_closed_halfline b)) by A5, JORDAN5A:24
.= (Cl (R^1 (left_open_halfline a))) \/ (Cl (R^1 (right_closed_halfline b))) by A4, JORDAN5A:24
.= (R^1 (left_closed_halfline a)) \/ (Cl (R^1 (right_closed_halfline b))) by A7, BORSUK_5:51, TOPREALB:def_3
.= (left_closed_halfline a) \/ (right_closed_halfline b) by A4, A7, PRE_TOPC:22 ;
hence  Fr X = {a,b} by A3, TOPS_1:def_2; ::_thesis: verum
end;

theorem Th35: :: RCOMP_3:35
for a, b being real number
for X being Subset of R^1 st a < b & X = ].a,b.] holds
Fr X = {a,b}
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st a < b & X = ].a,b.] holds
Fr X = {a,b}
let X be Subset of R^1; ::_thesis: ( a < b & X = ].a,b.] implies Fr X = {a,b} )
assume that
A1: a < b and
A2: X = ].a,b.] ; ::_thesis: Fr X = {a,b}
A3: ( Cl X = [.a,b.] & [.a,b.] /\ ((left_closed_halfline a) \/ (right_closed_halfline b)) = {a,b} ) by A1, A2, Th8, BORSUK_5:36;
A4: ].a,b.] ` = (left_closed_halfline a) \/ (right_open_halfline b) by XXREAL_1:399;
set RO = R^1 (right_open_halfline b);
set LC = R^1 (left_closed_halfline a);
A5: R^1 (right_open_halfline b) = right_open_halfline b by TOPREALB:def_3;
A6: R^1 (left_closed_halfline a) = left_closed_halfline a by TOPREALB:def_3;
Cl (X `) = Cl (].a,b.] `) by A2, JORDAN5A:24, TOPMETR:17
.= (Cl (left_closed_halfline a)) \/ (Cl (right_open_halfline b)) by A4, Th3
.= (Cl (R^1 (left_closed_halfline a))) \/ (Cl (right_open_halfline b)) by A6, JORDAN5A:24
.= (Cl (R^1 (left_closed_halfline a))) \/ (Cl (R^1 (right_open_halfline b))) by A5, JORDAN5A:24
.= (R^1 (left_closed_halfline a)) \/ (Cl (R^1 (right_open_halfline b))) by PRE_TOPC:22
.= (left_closed_halfline a) \/ (right_closed_halfline b) by A6, BORSUK_5:49, TOPREALB:def_3 ;
hence  Fr X = {a,b} by A3, TOPS_1:def_2; ::_thesis: verum
end;

theorem :: RCOMP_3:36
for a, b being real number
for X being Subset of R^1 st X = [.a,b.] holds
Int X = ].a,b.[
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st X = [.a,b.] holds
Int X = ].a,b.[
let X be Subset of R^1; ::_thesis: ( X = [.a,b.] implies Int X = ].a,b.[ )
assume A1: X = [.a,b.] ; ::_thesis: Int X = ].a,b.[
A2: Int X c= X by TOPS_1:16;
thus  Int X c= ].a,b.[ :: according to XBOOLE_0:def_10 ::_thesis: ].a,b.[ c= Int X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int X or x in ].a,b.[ )
assume A3: x in Int X ; ::_thesis: x in ].a,b.[
then reconsider x = x as Point of R^1 ;
A4: now__::_thesis:_(_not_x_=_a_&_not_x_=_b_)
now__::_thesis:_not_a_>_b
assume  a > b ; ::_thesis: contradiction
then  X = {} R^1 by A1, XXREAL_1:29;
hence  contradiction by A3; ::_thesis: verum
end;
then  Fr X = {a,b} by A1, Th32;
then A5: ( a in Fr X & b in Fr X ) by TARSKI:def_2;
A6: Int X misses Fr X by TOPS_1:39;
assume  ( x = a or x = b ) ; ::_thesis: contradiction
hence  contradiction by A3, A6, A5, XBOOLE_0:3; ::_thesis: verum
end;
 x <= b by A1, A2, A3, XXREAL_1:1;
then A7: x < b by A4, XXREAL_0:1;
 a <= x by A1, A2, A3, XXREAL_1:1;
then  a < x by A4, XXREAL_0:1;
hence  x in ].a,b.[ by A7, XXREAL_1:4; ::_thesis: verum
end;
reconsider Y = ].a,b.[ as open Subset of R^1 by BORSUK_5:39, TOPMETR:17;
 Y c= Int X by A1, TOPS_1:24, XXREAL_1:37;
hence  ].a,b.[ c= Int X ; ::_thesis: verum
end;

theorem :: RCOMP_3:37
for a, b being real number
for X being Subset of R^1 st X = ].a,b.[ holds
Int X = ].a,b.[
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st X = ].a,b.[ holds
Int X = ].a,b.[
let X be Subset of R^1; ::_thesis: ( X = ].a,b.[ implies Int X = ].a,b.[ )
assume A1: X = ].a,b.[ ; ::_thesis: Int X = ].a,b.[
then reconsider X = X as open Subset of R^1 by BORSUK_5:39;
 Int X = X by TOPS_1:23;
hence  Int X = ].a,b.[ by A1; ::_thesis: verum
end;

theorem Th38: :: RCOMP_3:38
for a, b being real number
for X being Subset of R^1 st X = [.a,b.[ holds
Int X = ].a,b.[
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st X = [.a,b.[ holds
Int X = ].a,b.[
let X be Subset of R^1; ::_thesis: ( X = [.a,b.[ implies Int X = ].a,b.[ )
assume A1: X = [.a,b.[ ; ::_thesis: Int X = ].a,b.[
A2: Int X c= X by TOPS_1:16;
thus  Int X c= ].a,b.[ :: according to XBOOLE_0:def_10 ::_thesis: ].a,b.[ c= Int X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int X or x in ].a,b.[ )
assume A3: x in Int X ; ::_thesis: x in ].a,b.[
then reconsider x = x as Point of R^1 ;
A4: now__::_thesis:_not_x_=_a
now__::_thesis:_not_a_>=_b
assume  a >= b ; ::_thesis: contradiction
then  X = {} R^1 by A1, XXREAL_1:27;
hence  contradiction by A3; ::_thesis: verum
end;
then  Fr X = {a,b} by A1, Th34;
then A5: ( Int X misses Fr X & a in Fr X ) by TARSKI:def_2, TOPS_1:39;
assume  x = a ; ::_thesis: contradiction
hence  contradiction by A3, A5, XBOOLE_0:3; ::_thesis: verum
end;
 a <= x by A1, A2, A3, XXREAL_1:3;
then A6: a < x by A4, XXREAL_0:1;
 x < b by A1, A2, A3, XXREAL_1:3;
hence  x in ].a,b.[ by A6, XXREAL_1:4; ::_thesis: verum
end;
reconsider Y = ].a,b.[ as open Subset of R^1 by BORSUK_5:39, TOPMETR:17;
 Y c= Int X by A1, TOPS_1:24, XXREAL_1:45;
hence  ].a,b.[ c= Int X ; ::_thesis: verum
end;

theorem Th39: :: RCOMP_3:39
for a, b being real number
for X being Subset of R^1 st X = ].a,b.] holds
Int X = ].a,b.[
proof
let a, b be real number ; ::_thesis: for X being Subset of R^1 st X = ].a,b.] holds
Int X = ].a,b.[
let X be Subset of R^1; ::_thesis: ( X = ].a,b.] implies Int X = ].a,b.[ )
assume A1: X = ].a,b.] ; ::_thesis: Int X = ].a,b.[
A2: Int X c= X by TOPS_1:16;
thus  Int X c= ].a,b.[ :: according to XBOOLE_0:def_10 ::_thesis: ].a,b.[ c= Int X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int X or x in ].a,b.[ )
assume A3: x in Int X ; ::_thesis: x in ].a,b.[
then reconsider x = x as Point of R^1 ;
A4: now__::_thesis:_not_x_=_b
now__::_thesis:_not_a_>=_b
assume  a >= b ; ::_thesis: contradiction
then  X = {} R^1 by A1, XXREAL_1:26;
hence  contradiction by A3; ::_thesis: verum
end;
then  Fr X = {a,b} by A1, Th35;
then A5: ( Int X misses Fr X & b in Fr X ) by TARSKI:def_2, TOPS_1:39;
assume  x = b ; ::_thesis: contradiction
hence  contradiction by A3, A5, XBOOLE_0:3; ::_thesis: verum
end;
 x <= b by A1, A2, A3, XXREAL_1:2;
then A6: x < b by A4, XXREAL_0:1;
 a < x by A1, A2, A3, XXREAL_1:2;
hence  x in ].a,b.[ by A6, XXREAL_1:4; ::_thesis: verum
end;
reconsider Y = ].a,b.[ as open Subset of R^1 by BORSUK_5:39, TOPMETR:17;
 Y c= Int X by A1, TOPS_1:24, XXREAL_1:41;
hence  ].a,b.[ c= Int X ; ::_thesis: verum
end;

registration
let T be real-membered TopStruct ;
let X be interval Subset of T;
clusterT | X ->  interval ;
coherence
 T | X is interval
proof
thus  [#] (T | X) is interval by PRE_TOPC:def_5; :: according to TOPALG_2:def_3 ::_thesis: verum
end;
end;

registration
let A be interval Subset of REAL;
cluster R^1 A ->  interval ;
coherence
 R^1 A is interval by TOPREALB:def_3;
end;

registration
cluster connected  ->  interval for Element of K32( the carrier of R^1);
coherence
 for b1 being Subset of R^1 st b1 is connected holds
b1 is interval
proof
let X be Subset of R^1; ::_thesis: ( X is connected implies X is interval )
assume A1: X is connected ; ::_thesis: X is interval
let a be ext-real number ; :: according to XXREAL_2:def_12 ::_thesis: for b1 being set holds
( not a in X or not b1 in X or [.a,b1.] c= X )
thus  for b1 being set holds
( not a in X or not b1 in X or [.a,b1.] c= X ) by A1, BORSUK_5:77; ::_thesis: verum
end;
cluster interval  ->  connected for Element of K32( the carrier of R^1);
coherence
 for b1 being Subset of R^1 st b1 is interval holds
b1 is connected
proof
let X be Subset of R^1; ::_thesis: ( X is interval implies X is connected )
assume A2: X is interval ; ::_thesis: X is connected
A3: X is Subset of REAL by MEMBERED:3;
percases ( X is empty or X = REAL or ex a being real number st X = left_closed_halfline a or ex a being real number st X = left_open_halfline a or ex a being real number st X = right_closed_halfline a or ex a being real number st X = right_open_halfline a or ex a, b being real number st
( a <= b & X = [.a,b.] ) or ex a, b being real number st
( a < b & X = [.a,b.[ ) or ex a, b being real number st
( a < b & X = ].a,b.] ) or ex a, b being real number st
( a < b & X = ].a,b.[ ) ) by A2, A3, Th29;
suppose X is empty ; ::_thesis: X is connected
then reconsider A = X as empty Subset of R^1 ;
 A is interval ;
hence  X is connected ; ::_thesis: verum
end;
suppose X = REAL ; ::_thesis: X is connected
then reconsider A = X as non empty interval Subset of R^1 ;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a being real number st X = left_closed_halfline a ; ::_thesis: X is connected
then consider a being real number  such that
A4: X = left_closed_halfline a ;
reconsider A = X as non empty interval Subset of R^1 by A4;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a being real number st X = left_open_halfline a ; ::_thesis: X is connected
then consider a being real number  such that
A5: X = left_open_halfline a ;
reconsider A = X as non empty interval Subset of R^1 by A5;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a being real number st X = right_closed_halfline a ; ::_thesis: X is connected
then consider a being real number  such that
A6: X = right_closed_halfline a ;
reconsider A = X as non empty interval Subset of R^1 by A6;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a being real number st X = right_open_halfline a ; ::_thesis: X is connected
then consider a being real number  such that
A7: X = right_open_halfline a ;
reconsider A = X as non empty interval Subset of R^1 by A7;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & X = [.a,b.] ) ; ::_thesis: X is connected
then reconsider A = X as non empty interval Subset of R^1 by XXREAL_1:1;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = [.a,b.[ ) ; ::_thesis: X is connected
then reconsider A = X as non empty interval Subset of R^1 by XXREAL_1:3;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.] ) ; ::_thesis: X is connected
then reconsider A = X as non empty interval Subset of R^1 by XXREAL_1:2;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & X = ].a,b.[ ) ; ::_thesis: X is connected
then reconsider A = X as non empty interval Subset of R^1 by XXREAL_1:33;
 R^1 | A is connected ;
hence  R^1 | X is connected ; :: according to CONNSP_1:def_3 ::_thesis: verum
end;
end;
end;
end;

begin

registration
let r be real number ;
cluster Closed-Interval-TSpace (r,r) -> 1 -element ;
coherence
 Closed-Interval-TSpace (r,r) is 1 -element
proof
 ( {r} is 1 -element & the carrier of (Closed-Interval-TSpace (r,r)) = [.r,r.] ) by TOPMETR:18;
hence  the carrier of (Closed-Interval-TSpace (r,r)) is 1 -element by XXREAL_1:17; :: according to STRUCT_0:def_19 ::_thesis: verum
end;
end;

theorem :: RCOMP_3:40
for r, s being real number st r <= s holds
for A being Subset of (Closed-Interval-TSpace (r,s)) holds A is real-bounded Subset of REAL
proof
let r, s be real number ; ::_thesis: ( r <= s implies for A being Subset of (Closed-Interval-TSpace (r,s)) holds A is real-bounded Subset of REAL )
assume  r <= s ; ::_thesis: for A being Subset of (Closed-Interval-TSpace (r,s)) holds A is real-bounded Subset of REAL
then A1: the carrier of (Closed-Interval-TSpace (r,s)) = [.r,s.] by TOPMETR:18;
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: A is real-bounded Subset of REAL
 ( A is bounded_above & A is bounded_below ) by A1, XXREAL_2:43, XXREAL_2:44;
hence  A is real-bounded Subset of REAL by A1, XBOOLE_1:1; ::_thesis: verum
end;

theorem Th41: :: RCOMP_3:41
for r, s, a, b being real number st r <= s holds
for X being Subset of (Closed-Interval-TSpace (r,s)) st X = [.a,b.[ & r < a & b <= s holds
Int X = ].a,b.[
proof
let r, s, a, b be real number ; ::_thesis: ( r <= s implies for X being Subset of (Closed-Interval-TSpace (r,s)) st X = [.a,b.[ & r < a & b <= s holds
Int X = ].a,b.[ )
set L = Closed-Interval-TSpace (r,s);
set c = (r + a) / 2;
set C1 = R^1 ].((r + a) / 2),b.[;
A1: R^1 ].((r + a) / 2),b.[ = ].((r + a) / 2),b.[ by TOPREALB:def_3;
assume  r <= s ; ::_thesis: for X being Subset of (Closed-Interval-TSpace (r,s)) st X = [.a,b.[ & r < a & b <= s holds
Int X = ].a,b.[
then A2: the carrier of (Closed-Interval-TSpace (r,s)) = [.r,s.] by TOPMETR:18;
let X be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( X = [.a,b.[ & r < a & b <= s implies Int X = ].a,b.[ )
assume that
A3: X = [.a,b.[ and
A4: r < a and
A5: b <= s ; ::_thesis: Int X = ].a,b.[
A6: r < (r + a) / 2 by A4, XREAL_1:226;
A7: R^1 ].((r + a) / 2),b.[ c= the carrier of (Closed-Interval-TSpace (r,s))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in R^1 ].((r + a) / 2),b.[ or x in the carrier of (Closed-Interval-TSpace (r,s)) )
assume A8: x in R^1 ].((r + a) / 2),b.[ ; ::_thesis: x in the carrier of (Closed-Interval-TSpace (r,s))
then reconsider x = x as Real by A1;
 x < b by A1, A8, XXREAL_1:4;
then A9: x <= s by A5, XXREAL_0:2;
 (r + a) / 2 < x by A1, A8, XXREAL_1:4;
then  r <= x by A6, XXREAL_0:2;
hence  x in the carrier of (Closed-Interval-TSpace (r,s)) by A2, A9, XXREAL_1:1; ::_thesis: verum
end;
reconsider A = X as Subset of R^1 by PRE_TOPC:11;
A10: (r + a) / 2 < a by A4, XREAL_1:226;
 A c= R^1 ].((r + a) / 2),b.[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in R^1 ].((r + a) / 2),b.[ )
assume A11: x in A ; ::_thesis: x in R^1 ].((r + a) / 2),b.[
then reconsider x = x as Real by A3;
 a <= x by A3, A11, XXREAL_1:3;
then A12: (r + a) / 2 < x by A10, XXREAL_0:2;
 x < b by A3, A11, XXREAL_1:3;
hence  x in R^1 ].((r + a) / 2),b.[ by A1, A12, XXREAL_1:4; ::_thesis: verum
end;
then  Int A = Int X by A7, TOPS_3:57;
hence  Int X = ].a,b.[ by A3, Th38; ::_thesis: verum
end;

theorem Th42: :: RCOMP_3:42
for r, s, a, b being real number st r <= s holds
for X being Subset of (Closed-Interval-TSpace (r,s)) st X = ].a,b.] & r <= a & b < s holds
Int X = ].a,b.[
proof
let r, s, a, b be real number ; ::_thesis: ( r <= s implies for X being Subset of (Closed-Interval-TSpace (r,s)) st X = ].a,b.] & r <= a & b < s holds
Int X = ].a,b.[ )
set L = Closed-Interval-TSpace (r,s);
set c = (b + s) / 2;
set C1 = R^1 ].a,((b + s) / 2).[;
A1: R^1 ].a,((b + s) / 2).[ = ].a,((b + s) / 2).[ by TOPREALB:def_3;
assume  r <= s ; ::_thesis: for X being Subset of (Closed-Interval-TSpace (r,s)) st X = ].a,b.] & r <= a & b < s holds
Int X = ].a,b.[
then A2: the carrier of (Closed-Interval-TSpace (r,s)) = [.r,s.] by TOPMETR:18;
let X be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( X = ].a,b.] & r <= a & b < s implies Int X = ].a,b.[ )
assume that
A3: X = ].a,b.] and
A4: r <= a and
A5: b < s ; ::_thesis: Int X = ].a,b.[
A6: (b + s) / 2 < s by A5, XREAL_1:226;
A7: R^1 ].a,((b + s) / 2).[ c= the carrier of (Closed-Interval-TSpace (r,s))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in R^1 ].a,((b + s) / 2).[ or x in the carrier of (Closed-Interval-TSpace (r,s)) )
assume A8: x in R^1 ].a,((b + s) / 2).[ ; ::_thesis: x in the carrier of (Closed-Interval-TSpace (r,s))
then reconsider x = x as Real by A1;
 x < (b + s) / 2 by A1, A8, XXREAL_1:4;
then A9: x <= s by A6, XXREAL_0:2;
 a < x by A1, A8, XXREAL_1:4;
then  r <= x by A4, XXREAL_0:2;
hence  x in the carrier of (Closed-Interval-TSpace (r,s)) by A2, A9, XXREAL_1:1; ::_thesis: verum
end;
reconsider A = X as Subset of R^1 by PRE_TOPC:11;
A10: b < (b + s) / 2 by A5, XREAL_1:226;
 A c= R^1 ].a,((b + s) / 2).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in R^1 ].a,((b + s) / 2).[ )
assume A11: x in A ; ::_thesis: x in R^1 ].a,((b + s) / 2).[
then reconsider x = x as Real by A3;
 x <= b by A3, A11, XXREAL_1:2;
then A12: x < (b + s) / 2 by A10, XXREAL_0:2;
 a < x by A3, A11, XXREAL_1:2;
hence  x in R^1 ].a,((b + s) / 2).[ by A1, A12, XXREAL_1:4; ::_thesis: verum
end;
then  Int A = Int X by A7, TOPS_3:57;
hence  Int X = ].a,b.[ by A3, Th39; ::_thesis: verum
end;

theorem Th43: :: RCOMP_3:43
for r, s being real number
for X being Subset of (Closed-Interval-TSpace (r,s))
for Y being Subset of REAL st X = Y holds
( X is connected iff Y is interval )
proof
let r, s be real number ; ::_thesis: for X being Subset of (Closed-Interval-TSpace (r,s))
for Y being Subset of REAL st X = Y holds
( X is connected iff Y is interval )
let X be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: for Y being Subset of REAL st X = Y holds
( X is connected iff Y is interval )
let Y be Subset of REAL; ::_thesis: ( X = Y implies ( X is connected iff Y is interval ) )
assume A1: X = Y ; ::_thesis: ( X is connected iff Y is interval )
reconsider Z = X as Subset of R^1 by A1, TOPMETR:17;
hereby  ::_thesis: ( Y is interval implies X is connected )
assume  X is connected ; ::_thesis: Y is interval
then  Z is connected by CONNSP_1:23;
hence  Y is interval by A1; ::_thesis: verum
end;
assume  Y is interval ; ::_thesis: X is connected
then  Z is connected by A1;
hence  X is connected by CONNSP_1:23; ::_thesis: verum
end;

registration
let T be TopSpace;
cluster open closed connected for Element of K32( the carrier of T);
existence
 ex b1 being Subset of T st
( b1 is open & b1 is closed & b1 is connected )
proof
take  {} T ; ::_thesis: ( {} T is open & {} T is closed & {} T is connected )
thus  ( {} T is open & {} T is closed & {} T is connected ) ; ::_thesis: verum
end;
end;

registration
let T be non empty connected TopSpace;
cluster non empty open closed connected for Element of K32( the carrier of T);
existence
 ex b1 being Subset of T st
( not b1 is empty & b1 is open & b1 is closed & b1 is connected )
proof
take  [#] T ; ::_thesis: ( not [#] T is empty & [#] T is open & [#] T is closed & [#] T is connected )
thus  ( not [#] T is empty & [#] T is open & [#] T is closed & [#] T is connected ) by CONNSP_1:27; ::_thesis: verum
end;
end;

theorem Th44: :: RCOMP_3:44
for r, s being real number st r <= s holds
for X being open connected Subset of (Closed-Interval-TSpace (r,s)) holds
( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
proof
let r, s be real number ; ::_thesis: ( r <= s implies for X being open connected Subset of (Closed-Interval-TSpace (r,s)) holds
( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) )
set L = Closed-Interval-TSpace (r,s);
assume A1: r <= s ; ::_thesis: for X being open connected Subset of (Closed-Interval-TSpace (r,s)) holds
( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
then A2: the carrier of (Closed-Interval-TSpace (r,s)) = [.r,s.] by TOPMETR:18;
let X be open connected Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
 X is bounded_below bounded_above Subset of REAL by A2, XBOOLE_1:1, XXREAL_2:43, XXREAL_2:44;
then reconsider Y = X as real-bounded interval Subset of REAL by Th43;
A3: ( the carrier of (Closed-Interval-TSpace (r,s)) = [#] (Closed-Interval-TSpace (r,s)) & Closed-Interval-TSpace (r,s) is connected ) by A1, TREAL_1:20;
A4: s in [.r,s.] by A1, XXREAL_1:1;
A5: r in [.r,s.] by A1, XXREAL_1:1;
percases ( Y is empty or Y = [#] REAL or ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a or ex a, b being real number st
( a <= b & Y = [.a,b.] ) or ex a, b being real number st
( a < b & Y = [.a,b.[ ) or ex a, b being real number st
( a < b & Y = ].a,b.] ) or ex a, b being real number st
( a < b & Y = ].a,b.[ ) ) by Th29;
suppose ( Y is empty or Y = [#] REAL ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
suppose ( ex a being real number st Y = left_closed_halfline a or ex a being real number st Y = left_open_halfline a or ex a being real number st Y = right_closed_halfline a or ex a being real number st Y = right_open_halfline a ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) ; ::_thesis: verum
end;
suppose ex a, b being real number st
( a <= b & Y = [.a,b.] ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
then consider a, b being real number  such that
A6: a <= b and
A7: Y = [.a,b.] ;
A8: X <> {} (Closed-Interval-TSpace (r,s)) by A6, A7, XXREAL_1:1;
A9: ( r <= a & b <= s ) by A2, A6, A7, XXREAL_1:50;
then A10: X is closed by A7, TOPREALA:23;
now__::_thesis:_(_not_r_<>_a_&_not_b_<>_s_)
assume A11: ( r <> a or b <> s ) ; ::_thesis: contradiction
percases ( r < a or b < s ) by A9, A11, XXREAL_0:1;
suppose r < a ; ::_thesis: contradiction
then  not r in X by A7, XXREAL_1:1;
hence  contradiction by A2, A3, A5, A8, A10, CONNSP_1:13; ::_thesis: verum
end;
suppose b < s ; ::_thesis: contradiction
then  not s in X by A7, XXREAL_1:1;
hence  contradiction by A2, A3, A4, A8, A10, CONNSP_1:13; ::_thesis: verum
end;
end;
end;
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) by A7; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = [.a,b.[ ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
then consider a, b being real number  such that
A12: a < b and
A13: Y = [.a,b.[ ;
A14: b <= s by A2, A12, A13, XXREAL_1:52;
A15: r <= a by A2, A12, A13, XXREAL_1:52;
now__::_thesis:_not_r_<>_a
assume  r <> a ; ::_thesis: contradiction
then A16: r < a by A15, XXREAL_0:1;
now__::_thesis:_not_Int_X_=_X
 Int X = ].a,b.[ by A1, A13, A14, A16, Th41;
then A17: not a in Int X by XXREAL_1:4;
assume  Int X = X ; ::_thesis: contradiction
hence  contradiction by A12, A13, A17, XXREAL_1:3; ::_thesis: verum
end;
hence  contradiction by TOPS_1:23; ::_thesis: verum
end;
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) by A12, A13, A14; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.] ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
then consider a, b being real number  such that
A18: a < b and
A19: Y = ].a,b.] ;
A20: r <= a by A2, A18, A19, XXREAL_1:53;
A21: b <= s by A2, A18, A19, XXREAL_1:53;
now__::_thesis:_not_b_<>_s
assume  b <> s ; ::_thesis: contradiction
then A22: b < s by A21, XXREAL_0:1;
now__::_thesis:_not_Int_X_=_X
 Int X = ].a,b.[ by A1, A19, A20, A22, Th42;
then A23: not b in Int X by XXREAL_1:4;
assume  Int X = X ; ::_thesis: contradiction
hence  contradiction by A18, A19, A23, XXREAL_1:2; ::_thesis: verum
end;
hence  contradiction by TOPS_1:23; ::_thesis: verum
end;
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) by A18, A19, A20; ::_thesis: verum
end;
suppose ex a, b being real number st
( a < b & Y = ].a,b.[ ) ; ::_thesis: ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) )
then consider a, b being real number  such that
A24: ( a < b & Y = ].a,b.[ ) ;
 ( r <= a & b <= s ) by A2, A24, XXREAL_1:51;
hence  ( X is empty or X = [.r,s.] or ex a being real number st
( r < a & a <= s & X = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & X = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & X = ].a,b.[ ) ) by A24; ::_thesis: verum
end;
end;
end;

begin

deffunc H1( set ) ->  set = $1;

defpred S1[ set , set ] means $1 c= $2;

theorem Th45: :: RCOMP_3:45
for T being 1-sorted
for F being finite Subset-Family of T
for F1 being Subset-Family of T st F is Cover of T & F1 = F \  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  holds
F1 is Cover of T
proof
let T be 1-sorted ; ::_thesis: for F being finite Subset-Family of T
for F1 being Subset-Family of T st F is Cover of T & F1 = F \  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  holds
F1 is Cover of T
let F be finite Subset-Family of T; ::_thesis: for F1 being Subset-Family of T st F is Cover of T & F1 = F \  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  holds
F1 is Cover of T
let F1 be Subset-Family of T; ::_thesis: ( F is Cover of T & F1 = F \  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  implies F1 is Cover of T )
assume that
A1: the carrier of T c= union F and
A2: F1 = F \  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  ; :: according to SETFAM_1:def_11 ::_thesis: F1 is Cover of T
set ZAW =  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  ;
thus  the carrier of T c= union F1 :: according to SETFAM_1:def_11 ::_thesis: verum
proof
let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in the carrier of T or t in union F1 )
assume  t in the carrier of T ; ::_thesis: t in union F1
then consider Z being set  such that
A3: t in Z and
A4: Z in F by A1, TARSKI:def_4;
set ALL =  {  X where X is Subset of T : ( Z c< X & X in F )  }  ;
percases (  {  X where X is Subset of T : ( Z c< X & X in F )  }  is empty or not  {  X where X is Subset of T : ( Z c< X & X in F )  }  is empty ) ;
supposeA5:  {  X where X is Subset of T : ( Z c< X & X in F )  }  is empty ; ::_thesis: t in union F1
now__::_thesis:_not_Z_in__{__X_where_X_is_Subset_of_T_:_(_X_in_F_&_ex_Y_being_Subset_of_T_st_
(_Y_in_F_&_X_c<_Y_)_)__}_
assume  Z in  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  ; ::_thesis: contradiction
then consider X being Subset of T such that
A6: Z = X and
 X in F and
A7: ex Y being Subset of T st
( Y in F & X c< Y ) ;
consider Y being Subset of T such that
A8: ( Y in F & X c< Y ) by A7;
 Y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A6, A8;
hence  contradiction by A5; ::_thesis: verum
end;
then  Z in F1 by A2, A4, XBOOLE_0:def_5;
hence  t in union F1 by A3, TARSKI:def_4; ::_thesis: verum
end;
suppose not  {  X where X is Subset of T : ( Z c< X & X in F )  }  is empty ; ::_thesis: t in union F1
then consider w being set  such that
A9: w in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by XBOOLE_0:def_1;
consider R being Relation of  {  X where X is Subset of T : ( Z c< X & X in F )  }   such that
A10: for x, y being set holds
( [x,y] in R iff ( x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & S1[x,y] ) ) from RELSET_1:sch_1();
A11: R is_reflexive_in  {  X where X is Subset of T : ( Z c< X & X in F )  }
proof
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or [x,x] in R )
thus  ( not x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or [x,x] in R ) by A10; ::_thesis: verum
end;
then A12: field R =  {  X where X is Subset of T : ( Z c< X & X in F )  }  by ORDERS_1:13;
A13: R partially_orders  {  X where X is Subset of T : ( Z c< X & X in F )  }
proof
thus  R is_reflexive_in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A11; :: according to ORDERS_1:def_7 ::_thesis: ( R is_transitive_in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & R is_antisymmetric_in  {  X where X is Subset of T : ( Z c< X & X in F )  }  )
thus  R is_transitive_in  {  X where X is Subset of T : ( Z c< X & X in F )  }  ::_thesis: R is_antisymmetric_in  {  X where X is Subset of T : ( Z c< X & X in F )  }
proof
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not z in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not [x,y] in R or not [y,z] in R or [x,z] in R )
assume that
A14: x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  and
 y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  and
A15: z in  {  X where X is Subset of T : ( Z c< X & X in F )  }  ; ::_thesis: ( not [x,y] in R or not [y,z] in R or [x,z] in R )
assume  ( [x,y] in R & [y,z] in R ) ; ::_thesis: [x,z] in R
then  ( x c= y & y c= z ) by A10;
then  x c= z by XBOOLE_1:1;
hence  [x,z] in R by A10, A14, A15; ::_thesis: verum
end;
let x, y be set ; :: according to RELAT_2:def_4 ::_thesis: ( not x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not [x,y] in R or not [y,x] in R or x = y )
assume that
 x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  and
 y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  ; ::_thesis: ( not [x,y] in R or not [y,x] in R or x = y )
assume  ( [x,y] in R & [y,x] in R ) ; ::_thesis: x = y
hence  ( x c= y & y c= x ) by A10; :: according to XBOOLE_0:def_10 ::_thesis: verum
end;
A16: R is reflexive by A11, A12, RELAT_2:def_9;
  {  X where X is Subset of T : ( Z c< X & X in F )  }  has_upper_Zorn_property_wrt R
proof
let Y be set ; :: according to ORDERS_1:def_9 ::_thesis: ( not Y c=  {  X where X is Subset of T : ( Z c< X & X in F )  }  or not R |_2 Y is being_linear-order or ex b1 being set st
( b1 in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b2 being set holds
( not b2 in Y or [b2,b1] in R ) ) ) )
assume that
A17: Y c=  {  X where X is Subset of T : ( Z c< X & X in F )  }  and
A18: R |_2 Y is being_linear-order ; ::_thesis: ex b1 being set st
( b1 in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b2 being set holds
( not b2 in Y or [b2,b1] in R ) ) )
percases ( not Y is empty or Y is empty ) ;
supposeA19: not Y is empty ; ::_thesis: ex b1 being set st
( b1 in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b2 being set holds
( not b2 in Y or [b2,b1] in R ) ) )
defpred S2[ set ] means ( not $1 is empty & $1 c= Y implies union $1 in Y );
take  union Y ; ::_thesis: ( union Y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b1 being set holds
( not b1 in Y or [b1,(union Y)] in R ) ) )
A20: S2[ {} ] ;
A21: for A, B being set st A in Y & B in Y holds
A \/ B in Y
proof
A22: R |_2 Y c= R by XBOOLE_1:17;
 R |_2 Y is connected by A18, ORDERS_1:def_5;
then A23: R |_2 Y is_connected_in field (R |_2 Y) by RELAT_2:def_14;
let A, B be set ; ::_thesis: ( A in Y & B in Y implies A \/ B in Y )
assume A24: ( A in Y & B in Y ) ; ::_thesis: A \/ B in Y
 field (R |_2 Y) = Y by A12, A16, A17, ORDERS_1:71;
then  ( [A,B] in R |_2 Y or [B,A] in R |_2 Y or A = B ) by A24, A23, RELAT_2:def_6;
then  ( A c= B or B c= A ) by A10, A22;
hence  A \/ B in Y by A24, XBOOLE_1:12; ::_thesis: verum
end;
A25: for x, B being set st x in Y & B c= Y & S2[B] holds
S2[B \/ {x}]
proof
let x, B be set ; ::_thesis: ( x in Y & B c= Y & S2[B] implies S2[B \/ {x}] )
assume that
A26: x in Y and
A27: ( B c= Y & S2[B] ) and
 not B \/ {x} is empty and
 B \/ {x} c= Y ; ::_thesis: union (B \/ {x}) in Y
A28: union {x} = x by ZFMISC_1:25;
percases ( B is empty or not B is empty ) ;
suppose B is empty ; ::_thesis: union (B \/ {x}) in Y
hence  union (B \/ {x}) in Y by A26, ZFMISC_1:25; ::_thesis: verum
end;
suppose not B is empty ; ::_thesis: union (B \/ {x}) in Y
then  x \/ (union B) in Y by A21, A26, A27;
hence  union (B \/ {x}) in Y by A28, ZFMISC_1:78; ::_thesis: verum
end;
end;
end;
consider y being set  such that
A29: y in Y by A19, XBOOLE_0:def_1;
 y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A17, A29;
then consider X being Subset of T such that
A30: X = y and
A31: Z c< X and
 X in F ;
A32: X c= union Y by A29, A30, ZFMISC_1:74;
then A33: Z <> union Y by A31, XBOOLE_0:def_8;
 Z c= X by A31, XBOOLE_0:def_8;
then  Z c= union Y by A32, XBOOLE_1:1;
then A34: Z c< union Y by A33, XBOOLE_0:def_8;
A35:  {  X where X is Subset of T : ( Z c< X & X in F )  }  c= F
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  or x in F )
assume  x in  {  X where X is Subset of T : ( Z c< X & X in F )  }  ; ::_thesis: x in F
then  ex X being Subset of T st
( X = x & Z c< X & X in F ) ;
hence  x in F ; ::_thesis: verum
end;
then A36: Y c= F by A17, XBOOLE_1:1;
A37: Y is finite by A17, A35;
 S2[Y] from FINSET_1:sch_2(A37, A20, A25);
then  union Y in F by A19, A36;
hence A38: union Y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A34; ::_thesis: for b1 being set holds
( not b1 in Y or [b1,(union Y)] in R )
let z be set ; ::_thesis: ( not z in Y or [z,(union Y)] in R )
assume A39: z in Y ; ::_thesis: [z,(union Y)] in R
then  S1[z, union Y] by ZFMISC_1:74;
hence  [z,(union Y)] in R by A10, A17, A38, A39; ::_thesis: verum
end;
supposeA40: Y is empty ; ::_thesis: ex b1 being set st
( b1 in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b2 being set holds
( not b2 in Y or [b2,b1] in R ) ) )
take  w ; ::_thesis: ( w in  {  X where X is Subset of T : ( Z c< X & X in F )  }  & ( for b1 being set holds
( not b1 in Y or [b1,w] in R ) ) )
thus  w in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A9; ::_thesis: for b1 being set holds
( not b1 in Y or [b1,w] in R )
thus  for b1 being set holds
( not b1 in Y or [b1,w] in R ) by A40; ::_thesis: verum
end;
end;
end;
then consider M being set  such that
A41: M is_maximal_in R by A12, A13, ORDERS_1:63;
A42: M in field R by A41, ORDERS_1:def_11;
then A43: ex X being Subset of T st
( X = M & Z c< X & X in F ) by A12;
now__::_thesis:_not_M_in__{__X_where_X_is_Subset_of_T_:_(_X_in_F_&_ex_Y_being_Subset_of_T_st_
(_Y_in_F_&_X_c<_Y_)_)__}_
assume  M in  {  X where X is Subset of T : ( X in F & ex Y being Subset of T st
( Y in F & X c< Y ) )  }  ; ::_thesis: contradiction
then consider H being Subset of T such that
A44: M = H and
 H in F and
A45: ex Y being Subset of T st
( Y in F & H c< Y ) ;
consider Y being Subset of T such that
A46: Y in F and
A47: H c< Y by A45;
 Z c< Y by A43, A44, A47, XBOOLE_1:56;
then A48: Y in  {  X where X is Subset of T : ( Z c< X & X in F )  }  by A46;
 H c= Y by A47, XBOOLE_0:def_8;
then  [M,Y] in R by A10, A12, A42, A44, A48;
hence  contradiction by A12, A41, A44, A47, A48, ORDERS_1:def_11; ::_thesis: verum
end;
then A49: M in F1 by A2, A43, XBOOLE_0:def_5;
 Z c= M by A43, XBOOLE_0:def_8;
hence  t in union F1 by A3, A49, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
end;

theorem Th46: :: RCOMP_3:46
for S being 1 -element 1-sorted
for s being Point of S
for F being Subset-Family of S st F is Cover of S holds
{s} in F
proof
let S be 1 -element 1-sorted ; ::_thesis: for s being Point of S
for F being Subset-Family of S st F is Cover of S holds
{s} in F
let s be Point of S; ::_thesis: for F being Subset-Family of S st F is Cover of S holds
{s} in F
let F be Subset-Family of S; ::_thesis: ( F is Cover of S implies {s} in F )
assume A1: the carrier of S c= union F ; :: according to SETFAM_1:def_11 ::_thesis: {s} in F
consider d being Element of S such that
A2: the carrier of S = {d} by TEX_1:def_1;
 s in the carrier of S ;
then consider Z being set  such that
A3: s in Z and
A4: Z in F by A1, TARSKI:def_4;
A5: s = d by ZFMISC_1:def_10;
 Z = {s}
proof
thus  for x being set st x in Z holds
x in {s} by A4, A2, A5; :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {s} c= Z
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {s} or x in Z )
thus  ( not x in {s} or x in Z ) by A3, TARSKI:def_1; ::_thesis: verum
end;
hence  {s} in F by A4; ::_thesis: verum
end;

definition
let T be TopStruct ;
let F be Subset-Family of T;
attrF is connected  means :Def1: :: RCOMP_3:def 1
 for X being Subset of T st X in F holds
X is connected ;
end;

:: deftheorem Def1 defines connected RCOMP_3:def_1_:_
 for T being TopStruct
for F being Subset-Family of T holds
( F is connected iff for X being Subset of T st X in F holds
X is connected );

registration
let T be TopSpace;
cluster non empty open closed connected for Element of K32(K32( the carrier of T));
existence
 ex b1 being Subset-Family of T st
( not b1 is empty & b1 is open & b1 is closed & b1 is connected )
proof
 {({} T)} c= bool the carrier of T
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {({} T)} or x in bool the carrier of T )
assume  x in {({} T)} ; ::_thesis: x in bool the carrier of T
then  x = {} T by TARSKI:def_1;
hence  x in bool the carrier of T ; ::_thesis: verum
end;
then reconsider F = {({} T)} as Subset-Family of T ;
take  F ; ::_thesis: ( not F is empty & F is open & F is closed & F is connected )
thus  not F is empty ; ::_thesis: ( F is open & F is closed & F is connected )
thus  for P being Subset of T st P in F holds
P is open by TARSKI:def_1; :: according to TOPS_2:def_1 ::_thesis: ( F is closed & F is connected )
thus  for P being Subset of T st P in F holds
P is closed by TARSKI:def_1; :: according to TOPS_2:def_2 ::_thesis: F is connected
thus  for P being Subset of T st P in F holds
P is connected by TARSKI:def_1; :: according to RCOMP_3:def_1 ::_thesis: verum
end;
end;

   Lm3:  for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s)) st [.r,s.] in F & r <= s holds
( rng <*[.r,s.]*> c= F & union (rng <*[.r,s.]*>) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) ) )
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s)) st [.r,s.] in F & r <= s holds
( rng <*[.r,s.]*> c= F & union (rng <*[.r,s.]*>) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) ) )
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: ( [.r,s.] in F & r <= s implies ( rng <*[.r,s.]*> c= F & union (rng <*[.r,s.]*>) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) ) ) )
assume that
A1: [.r,s.] in F and
A2: r <= s ; ::_thesis: ( rng <*[.r,s.]*> c= F & union (rng <*[.r,s.]*>) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) ) )
set f = <*[.r,s.]*>;
A3: rng <*[.r,s.]*> = {[.r,s.]} by FINSEQ_1:38;
thus  rng <*[.r,s.]*> c= F ::_thesis: ( union (rng <*[.r,s.]*>) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) ) )
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng <*[.r,s.]*> or a in F )
assume  a in rng <*[.r,s.]*> ; ::_thesis: a in F
hence  a in F by A1, A3, TARSKI:def_1; ::_thesis: verum
end;
thus  union (rng <*[.r,s.]*>) = [.r,s.] by A3, ZFMISC_1:25; ::_thesis: for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) )
let n be Nat; ::_thesis: ( 1 <= n implies ( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) )
assume A4: 1 <= n ; ::_thesis: ( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) )
hereby  ::_thesis: ( ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) )
assume  n <= len <*[.r,s.]*> ; ::_thesis: not <*[.r,s.]*> /. n is empty
then  n <= 1 by FINSEQ_1:39;
then  n = 1 by A4, XXREAL_0:1;
then  <*[.r,s.]*> /. n = [.r,s.] by FINSEQ_4:16;
hence  not <*[.r,s.]*> /. n is empty by A2, XXREAL_1:1; ::_thesis: verum
end;
hereby  ::_thesis: ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) )
assume  n + 1 <= len <*[.r,s.]*> ; ::_thesis: ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) )
then  n + 1 <= 0 + 1 by FINSEQ_1:39;
hence  ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) by A4, XREAL_1:6; ::_thesis: verum
end;
assume  n + 2 <= len <*[.r,s.]*> ; ::_thesis: upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2))
then  (n + 1) + 1 <= 0 + 1 by FINSEQ_1:39;
hence  upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) by XREAL_1:6; ::_thesis: verum
end;

theorem Th47: :: RCOMP_3:47
for L being TopSpace
for G, G1 being Subset-Family of L st G is Cover of L & G is finite holds
for ALL being set st G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  & ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  holds
ALL has_lower_Zorn_property_wrt RelIncl ALL
proof
let L be TopSpace; ::_thesis: for G, G1 being Subset-Family of L st G is Cover of L & G is finite holds
for ALL being set st G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  & ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  holds
ALL has_lower_Zorn_property_wrt RelIncl ALL
let G, G1 be Subset-Family of L; ::_thesis: ( G is Cover of L & G is finite implies for ALL being set st G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  & ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  holds
ALL has_lower_Zorn_property_wrt RelIncl ALL )
assume that
A1: G is Cover of L and
A2: G is finite ; ::_thesis: for ALL being set st G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  & ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  holds
ALL has_lower_Zorn_property_wrt RelIncl ALL
let ALL be set ; ::_thesis: ( G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  & ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  implies ALL has_lower_Zorn_property_wrt RelIncl ALL )
set ZAW =  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  ;
assume that
A3: G1 = G \  {  X where X is Subset of L : ( X in G & ex Y being Subset of L st
( Y in G & X c< Y ) )  }  and
A4: ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G1 )  }  ; ::_thesis: ALL has_lower_Zorn_property_wrt RelIncl ALL
A5: G1 is Cover of L by A1, A2, A3, Th45;
set R = RelIncl ALL;
A6: field (RelIncl ALL) = ALL by WELLORD2:def_1;
let Y be set ; :: according to ORDERS_1:def_10 ::_thesis: ( not Y c= ALL or not (RelIncl ALL) |_2 Y is being_linear-order or ex b1 being set st
( b1 in ALL & ( for b2 being set holds
( not b2 in Y or [b1,b2] in RelIncl ALL ) ) ) )
assume that
A7: Y c= ALL and
A8: (RelIncl ALL) |_2 Y is being_linear-order ; ::_thesis: ex b1 being set st
( b1 in ALL & ( for b2 being set holds
( not b2 in Y or [b1,b2] in RelIncl ALL ) ) )
percases ( not Y is empty or Y is empty ) ;
supposeA9: not Y is empty ; ::_thesis: ex b1 being set st
( b1 in ALL & ( for b2 being set holds
( not b2 in Y or [b1,b2] in RelIncl ALL ) ) )
defpred S2[ set ] means ( not $1 is empty implies meet $1 in Y );
set E =  {  H1(D) where D is Subset-Family of L : D in bool G1  }  ;
take x = meet Y; ::_thesis: ( x in ALL & ( for b1 being set holds
( not b1 in Y or [x,b1] in RelIncl ALL ) ) )
A10: ALL c=  {  H1(D) where D is Subset-Family of L : D in bool G1  }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in ALL or a in  {  H1(D) where D is Subset-Family of L : D in bool G1  }  )
assume  a in ALL ; ::_thesis: a in  {  H1(D) where D is Subset-Family of L : D in bool G1  }
then  ex C being Subset-Family of L st
( a = C & C is Cover of L & C c= G1 ) by A4;
hence  a in  {  H1(D) where D is Subset-Family of L : D in bool G1  }  ; ::_thesis: verum
end;
A11: bool G1 is finite by A2, A3;
  {  H1(D) where D is Subset-Family of L : D in bool G1  }  is finite from FRAENKEL:sch_21(A11);
then A12: Y is finite by A7, A10;
A13: for x, B being set st x in Y & B c= Y & S2[B] holds
S2[B \/ {x}]
proof
let x, B be set ; ::_thesis: ( x in Y & B c= Y & S2[B] implies S2[B \/ {x}] )
assume that
A14: x in Y and
 B c= Y and
A15: S2[B] and
 not B \/ {x} is empty ; ::_thesis: meet (B \/ {x}) in Y
percases ( B is empty or not B is empty ) ;
suppose B is empty ; ::_thesis: meet (B \/ {x}) in Y
hence  meet (B \/ {x}) in Y by A14, SETFAM_1:10; ::_thesis: verum
end;
supposeA16: not B is empty ; ::_thesis: meet (B \/ {x}) in Y
 (RelIncl ALL) |_2 Y is connected by A8, ORDERS_1:def_5;
then A17: (RelIncl ALL) |_2 Y is_connected_in field ((RelIncl ALL) |_2 Y) by RELAT_2:def_14;
 field ((RelIncl ALL) |_2 Y) = Y by A6, A7, ORDERS_1:71;
then  ( [x,(meet B)] in (RelIncl ALL) |_2 Y or [(meet B),x] in (RelIncl ALL) |_2 Y or x = meet B ) by A14, A15, A16, A17, RELAT_2:def_6;
then  ( [x,(meet B)] in RelIncl ALL or [(meet B),x] in RelIncl ALL or x = meet B ) by XBOOLE_0:def_4;
then A18: ( meet B c= x or x c= meet B ) by A7, A14, A15, A16, WELLORD2:def_1;
meet (B \/ {x}) = (meet B) /\ (meet {x}) by A16, SETFAM_1:9
.= (meet B) /\ x by SETFAM_1:10 ;
hence  meet (B \/ {x}) in Y by A14, A15, A16, A18, XBOOLE_1:28; ::_thesis: verum
end;
end;
end;
consider y being set  such that
A19: y in Y by A9, XBOOLE_0:def_1;
 y in ALL by A7, A19;
then A20: ex C being Subset-Family of L st
( y = C & C is Cover of L & C c= G1 ) by A4;
then reconsider X = x as Subset-Family of L by A19, SETFAM_1:7;
A21: S2[ {} ] ;
A22: S2[Y] from FINSET_1:sch_2(A12, A21, A13);
A23: X is Cover of L
proof
let a be set ; :: according to TARSKI:def_3,SETFAM_1:def_11 ::_thesis: ( not a in the carrier of L or a in union X )
assume A24: a in the carrier of L ; ::_thesis: a in union X
 meet Y in ALL by A7, A9, A22;
then consider C being Subset-Family of L such that
A25: meet Y = C and
A26: C is Cover of L and
 C c= G1 by A4;
 the carrier of L c= union C by A26, SETFAM_1:def_11;
hence  a in union X by A24, A25; ::_thesis: verum
end;
 x c= G1 by A19, A20, SETFAM_1:7;
hence A27: x in ALL by A4, A23; ::_thesis: for b1 being set holds
( not b1 in Y or [x,b1] in RelIncl ALL )
let y be set ; ::_thesis: ( not y in Y or [x,y] in RelIncl ALL )
assume A28: y in Y ; ::_thesis: [x,y] in RelIncl ALL
then  x c= y by SETFAM_1:7;
hence  [x,y] in RelIncl ALL by A7, A27, A28, WELLORD2:def_1; ::_thesis: verum
end;
supposeA29: Y is empty ; ::_thesis: ex b1 being set st
( b1 in ALL & ( for b2 being set holds
( not b2 in Y or [b1,b2] in RelIncl ALL ) ) )
take  G1 ; ::_thesis: ( G1 in ALL & ( for b1 being set holds
( not b1 in Y or [G1,b1] in RelIncl ALL ) ) )
thus  ( G1 in ALL & ( for b1 being set holds
( not b1 in Y or [G1,b1] in RelIncl ALL ) ) ) by A4, A5, A29; ::_thesis: verum
end;
end;
end;

theorem Th48: :: RCOMP_3:48
for L being TopSpace
for G, ALL being set st ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G )  }  holds
for M being set st M is_minimal_in RelIncl ALL & M in field (RelIncl ALL) holds
for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 )
proof
let L be TopSpace; ::_thesis: for G, ALL being set st ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G )  }  holds
for M being set st M is_minimal_in RelIncl ALL & M in field (RelIncl ALL) holds
for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 )
let G, ALL be set ; ::_thesis: ( ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G )  }  implies for M being set st M is_minimal_in RelIncl ALL & M in field (RelIncl ALL) holds
for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 ) )
assume A1: ALL =  {  C where C is Subset-Family of L : ( C is Cover of L & C c= G )  }  ; ::_thesis: for M being set st M is_minimal_in RelIncl ALL & M in field (RelIncl ALL) holds
for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 )
set R = RelIncl ALL;
let M be set ; ::_thesis: ( M is_minimal_in RelIncl ALL & M in field (RelIncl ALL) implies for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 ) )
assume that
A2: M is_minimal_in RelIncl ALL and
A3: M in field (RelIncl ALL) ; ::_thesis: for A1 being Subset of L st A1 in M holds
for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 )
A4: field (RelIncl ALL) = ALL by WELLORD2:def_1;
then consider C being Subset-Family of L such that
A5: M = C and
A6: C is Cover of L and
A7: C c= G by A1, A3;
let A1 be Subset of L; ::_thesis: ( A1 in M implies for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 ) )
assume A8: A1 in M ; ::_thesis: for A2, A3 being Subset of L holds
( not A2 in M or not A3 in M or not A1 c= A2 \/ A3 or not A1 <> A2 or not A1 <> A3 )
set Y = C \ {A1};
A9: C \ {A1} <> M by A8, ZFMISC_1:56;
given A2, A3 being Subset of L such that A10: A2 in M and
A11: A3 in M and
A12: A1 c= A2 \/ A3 and
A13: A1 <> A2 and
A14: A1 <> A3 ; ::_thesis: contradiction
A15: union C = [#] L by A6, SETFAM_1:45;
 union (C \ {A1}) = the carrier of L
proof
thus  union (C \ {A1}) c= the carrier of L ; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of L c= union (C \ {A1})
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of L or x in union (C \ {A1}) )
assume A16: x in the carrier of L ; ::_thesis: x in union (C \ {A1})
percases ( x in A1 or not x in A1 ) ;
supposeA17: x in A1 ; ::_thesis: x in union (C \ {A1})
percases ( x in A2 or x in A3 ) by A12, A17, XBOOLE_0:def_3;
supposeA18: x in A2 ; ::_thesis: x in union (C \ {A1})
 A2 in C \ {A1} by A5, A10, A13, ZFMISC_1:56;
hence  x in union (C \ {A1}) by A18, TARSKI:def_4; ::_thesis: verum
end;
supposeA19: x in A3 ; ::_thesis: x in union (C \ {A1})
 A3 in C \ {A1} by A5, A11, A14, ZFMISC_1:56;
hence  x in union (C \ {A1}) by A19, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
supposeA20: not x in A1 ; ::_thesis: x in union (C \ {A1})
consider Z being set  such that
A21: x in Z and
A22: Z in C by A15, A16, TARSKI:def_4;
 Z in C \ {A1} by A20, A21, A22, ZFMISC_1:56;
hence  x in union (C \ {A1}) by A21, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
then A23: C \ {A1} is Cover of L by SETFAM_1:def_11;
A24: C \ {A1} c= M by A5, XBOOLE_1:36;
then  C \ {A1} c= G by A5, A7, XBOOLE_1:1;
then A25: C \ {A1} in ALL by A1, A23;
then  [(C \ {A1}),M] in RelIncl ALL by A4, A3, A24, WELLORD2:def_1;
hence  contradiction by A4, A2, A9, A25, ORDERS_1:def_12; ::_thesis: verum
end;

definition
let r, s be real number ;
let F be Subset-Family of (Closed-Interval-TSpace (r,s));
assume that
B1: F is Cover of (Closed-Interval-TSpace (r,s)) and
B2: F is open and
B3: F is connected and
B4: r <= s ;
mode IntervalCover of F ->  FinSequence of bool REAL means :Def2: :: RCOMP_3:def 2
( rng it c= F & union (rng it) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len it implies not it /. n is empty ) & ( n + 1 <= len it implies ( lower_bound (it /. n) <= lower_bound (it /. (n + 1)) & upper_bound (it /. n) <= upper_bound (it /. (n + 1)) & lower_bound (it /. (n + 1)) < upper_bound (it /. n) ) ) & ( n + 2 <= len it implies upper_bound (it /. n) <= lower_bound (it /. (n + 2)) ) ) ) & ( [.r,s.] in F implies it = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & it . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & it . (len it) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len it holds
ex p, q being real number st
( r <= p & p < q & q <= s & it . n = ].p,q.[ ) ) ) ) );
existence
 ex b1 being FinSequence of bool REAL st
( rng b1 c= F & union (rng b1) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len b1 implies not b1 /. n is empty ) & ( n + 1 <= len b1 implies ( lower_bound (b1 /. n) <= lower_bound (b1 /. (n + 1)) & upper_bound (b1 /. n) <= upper_bound (b1 /. (n + 1)) & lower_bound (b1 /. (n + 1)) < upper_bound (b1 /. n) ) ) & ( n + 2 <= len b1 implies upper_bound (b1 /. n) <= lower_bound (b1 /. (n + 2)) ) ) ) & ( [.r,s.] in F implies b1 = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & b1 . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & b1 . (len b1) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len b1 holds
ex p, q being real number st
( r <= p & p < q & q <= s & b1 . n = ].p,q.[ ) ) ) ) )
proof
percases ( [.r,s.] in F or not [.r,s.] in F ) ;
supposeA1: [.r,s.] in F ; ::_thesis: ex b1 being FinSequence of bool REAL st
( rng b1 c= F & union (rng b1) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len b1 implies not b1 /. n is empty ) & ( n + 1 <= len b1 implies ( lower_bound (b1 /. n) <= lower_bound (b1 /. (n + 1)) & upper_bound (b1 /. n) <= upper_bound (b1 /. (n + 1)) & lower_bound (b1 /. (n + 1)) < upper_bound (b1 /. n) ) ) & ( n + 2 <= len b1 implies upper_bound (b1 /. n) <= lower_bound (b1 /. (n + 2)) ) ) ) & ( [.r,s.] in F implies b1 = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & b1 . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & b1 . (len b1) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len b1 holds
ex p, q being real number st
( r <= p & p < q & q <= s & b1 . n = ].p,q.[ ) ) ) ) )
take f = <*[.r,s.]*>; ::_thesis: ( rng f c= F & union (rng f) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len f implies not f /. n is empty ) & ( n + 1 <= len f implies ( lower_bound (f /. n) <= lower_bound (f /. (n + 1)) & upper_bound (f /. n) <= upper_bound (f /. (n + 1)) & lower_bound (f /. (n + 1)) < upper_bound (f /. n) ) ) & ( n + 2 <= len f implies upper_bound (f /. n) <= lower_bound (f /. (n + 2)) ) ) ) & ( [.r,s.] in F implies f = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & f . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & f . (len f) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len f holds
ex p, q being real number st
( r <= p & p < q & q <= s & f . n = ].p,q.[ ) ) ) ) )
A2: rng f = {[.r,s.]} by FINSEQ_1:38;
thus  rng f c= F ::_thesis: ( union (rng f) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len f implies not f /. n is empty ) & ( n + 1 <= len f implies ( lower_bound (f /. n) <= lower_bound (f /. (n + 1)) & upper_bound (f /. n) <= upper_bound (f /. (n + 1)) & lower_bound (f /. (n + 1)) < upper_bound (f /. n) ) ) & ( n + 2 <= len f implies upper_bound (f /. n) <= lower_bound (f /. (n + 2)) ) ) ) & ( [.r,s.] in F implies f = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & f . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & f . (len f) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len f holds
ex p, q being real number st
( r <= p & p < q & q <= s & f . n = ].p,q.[ ) ) ) ) )
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in rng f or a in F )
assume  a in rng f ; ::_thesis: a in F
hence  a in F by A1, A2, TARSKI:def_1; ::_thesis: verum
end;
thus  union (rng f) = [.r,s.] by A2, ZFMISC_1:25; ::_thesis: ( ( for n being Nat st 1 <= n holds
( ( n <= len f implies not f /. n is empty ) & ( n + 1 <= len f implies ( lower_bound (f /. n) <= lower_bound (f /. (n + 1)) & upper_bound (f /. n) <= upper_bound (f /. (n + 1)) & lower_bound (f /. (n + 1)) < upper_bound (f /. n) ) ) & ( n + 2 <= len f implies upper_bound (f /. n) <= lower_bound (f /. (n + 2)) ) ) ) & ( [.r,s.] in F implies f = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & f . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & f . (len f) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len f holds
ex p, q being real number st
( r <= p & p < q & q <= s & f . n = ].p,q.[ ) ) ) ) )
thus  ( ( for n being Nat st 1 <= n holds
( ( n <= len f implies not f /. n is empty ) & ( n + 1 <= len f implies ( lower_bound (f /. n) <= lower_bound (f /. (n + 1)) & upper_bound (f /. n) <= upper_bound (f /. (n + 1)) & lower_bound (f /. (n + 1)) < upper_bound (f /. n) ) ) & ( n + 2 <= len f implies upper_bound (f /. n) <= lower_bound (f /. (n + 2)) ) ) ) & ( [.r,s.] in F implies f = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & f . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & f . (len f) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len f holds
ex p, q being real number st
( r <= p & p < q & q <= s & f . n = ].p,q.[ ) ) ) ) ) by B4, A1, Lm3; ::_thesis: verum
end;
supposeA3: not [.r,s.] in F ; ::_thesis: ex b1 being FinSequence of bool REAL st
( rng b1 c= F & union (rng b1) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len b1 implies not b1 /. n is empty ) & ( n + 1 <= len b1 implies ( lower_bound (b1 /. n) <= lower_bound (b1 /. (n + 1)) & upper_bound (b1 /. n) <= upper_bound (b1 /. (n + 1)) & lower_bound (b1 /. (n + 1)) < upper_bound (b1 /. n) ) ) & ( n + 2 <= len b1 implies upper_bound (b1 /. n) <= lower_bound (b1 /. (n + 2)) ) ) ) & ( [.r,s.] in F implies b1 = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & b1 . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & b1 . (len b1) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len b1 holds
ex p, q being real number st
( r <= p & p < q & q <= s & b1 . n = ].p,q.[ ) ) ) ) )
set L = Closed-Interval-TSpace (r,s);
A4: the carrier of (Closed-Interval-TSpace (r,s)) = [.r,s.] by B4, TOPMETR:18;
A5: now__::_thesis:_for_A_being_Subset_of_(Closed-Interval-TSpace_(r,s))_holds_
(_A_is_bounded_above_&_A_is_bounded_below_&_A_is_real-bounded_Subset_of_REAL_)
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( A is bounded_above & A is bounded_below & A is real-bounded Subset of REAL )
thus  ( A is bounded_above & A is bounded_below ) by A4, XXREAL_2:43, XXREAL_2:44; ::_thesis: A is real-bounded Subset of REAL
hence  A is real-bounded Subset of REAL by A4, XBOOLE_1:1; ::_thesis: verum
end;
 Closed-Interval-TSpace (r,s) is compact by B4, HEINE:4;
then  [#] (Closed-Interval-TSpace (r,s)) is compact by COMPTS_1:1;
then consider G being Subset-Family of (Closed-Interval-TSpace (r,s)) such that
A6: G c= F and
A7: G is Cover of [#] (Closed-Interval-TSpace (r,s)) and
A8: G is finite by B1, B2, COMPTS_1:def_4;
set ZAW =  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ;
set G1 = G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ;
set ALL =  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  ;
set R = RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  ;
A9: RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  is_antisymmetric_in  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by WELLORD2:21;
set RM =  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ;
set LM =  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ;
A10: G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  c= G by XBOOLE_1:36;
then A11: G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  c= F by A6, XBOOLE_1:1;
A12: for X being set st X in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  holds
X is interval Subset of REAL
proof
let X be set ; ::_thesis: ( X in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  implies X is interval Subset of REAL )
assume  X in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ; ::_thesis: X is interval Subset of REAL
then reconsider X = X as connected Subset of (Closed-Interval-TSpace (r,s)) by B3, A11, Def1;
reconsider Y = X as Subset of REAL by A4, XBOOLE_1:1;
 Y is interval by Th43;
hence  X is interval Subset of REAL ; ::_thesis: verum
end;
reconsider T = Closed-Interval-TSpace (r,s) as non empty connected TopSpace by B4, TREAL_1:20;
set LM1 =  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ;
A13:  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  c=  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  or x in  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  )
assume  x in  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: x in  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }
then  ex b being Real st
( x = upper_bound [.r,b.[ & [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) ;
hence  x in  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: verum
end;
A14:  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  or x in REAL )
assume  x in  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: x in REAL
then  ex b being Real st
( x = upper_bound [.r,b.[ & [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) ;
hence  x in REAL ; ::_thesis: verum
end;
set RM1 =  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ;
A15:  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  c=  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  or x in  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  )
assume  x in  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: x in  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }
then  ex b being Real st
( x = lower_bound ].b,s.] & ].b,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) ;
hence  x in  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: verum
end;
A16:  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  c= REAL
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  or x in REAL )
assume  x in  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  ; ::_thesis: x in REAL
then  ex b being Real st
( x = lower_bound ].b,s.] & ].b,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) ;
hence  x in REAL ; ::_thesis: verum
end;
A17: field (RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  ) =  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by WELLORD2:def_1;
 ( RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  is_reflexive_in  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  & RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  is_transitive_in  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  ) by WELLORD2:19, WELLORD2:20;
then  RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  partially_orders  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by A9, ORDERS_1:def_7;
then consider M being set  such that
A18: M is_minimal_in RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by A7, A8, A17, Th47, ORDERS_1:64;
A19: M in field (RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  ) by A18, ORDERS_1:def_12;
then consider C being Subset-Family of (Closed-Interval-TSpace (r,s)) such that
A20: M = C and
A21: C is Cover of (Closed-Interval-TSpace (r,s)) and
A22: C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A17;
A23: union C = [#] (Closed-Interval-TSpace (r,s)) by A21, SETFAM_1:45;
A24: s in [.r,s.] by B4, XXREAL_1:1;
then consider R2 being set  such that
A25: s in R2 and
A26: R2 in C by A4, A23, TARSKI:def_4;
 r in [.r,s.] by B4, XXREAL_1:1;
then consider R1 being set  such that
A27: r in R1 and
A28: R1 in C by A4, A23, TARSKI:def_4;
A29: R1 in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A22, A28;
then A30: R1 in F by A11;
A31: R2 in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A22, A26;
then A32: R2 in F by A11;
reconsider R1 = R1, R2 = R2 as open connected Subset of (Closed-Interval-TSpace (r,s)) by B2, B3, A11, A29, A31, Def1, TOPS_2:def_1;
A33: now__::_thesis:_not__{__(lower_bound_].c,s.])_where_c_is_Real_:_].c,s.]_in_G_\__{__X_where_X_is_Subset_of_(Closed-Interval-TSpace_(r,s))_:_(_X_in_G_&_ex_Y_being_Subset_of_(Closed-Interval-TSpace_(r,s))_st_
(_Y_in_G_&_X_c<_Y_)_)__}___}__is_empty
percases ( ex a being real number st
( r < a & a <= s & R2 = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & R2 = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & R2 = ].a,b.[ ) ) by B4, A3, A25, A32, Th44;
suppose ex a being real number st
( r < a & a <= s & R2 = [.r,a.[ ) ; ::_thesis: not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
hence  not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty by A25, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a being real number st
( r <= a & a < s & R2 = ].a,s.] ) ; ::_thesis: not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
then consider a being real number  such that
 r <= a and
 a < s and
A34: R2 = ].a,s.] ;
 a is Real by XREAL_0:def_1;
then  lower_bound ].a,s.] in  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  by A22, A26, A34;
hence  not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty ; ::_thesis: verum
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & R2 = ].a,b.[ ) ; ::_thesis: not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
hence  not  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty by A25, XXREAL_1:4; ::_thesis: verum
end;
end;
end;
A35: now__::_thesis:_not__{__(upper_bound_[.r,b.[)_where_b_is_Real_:_[.r,b.[_in_G_\__{__X_where_X_is_Subset_of_(Closed-Interval-TSpace_(r,s))_:_(_X_in_G_&_ex_Y_being_Subset_of_(Closed-Interval-TSpace_(r,s))_st_
(_Y_in_G_&_X_c<_Y_)_)__}___}__is_empty
percases ( ex a being real number st
( r < a & a <= s & R1 = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & R1 = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & R1 = ].a,b.[ ) ) by B4, A3, A27, A30, Th44;
suppose ex a being real number st
( r < a & a <= s & R1 = [.r,a.[ ) ; ::_thesis: not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
then consider a being real number  such that
 r < a and
 a <= s and
A36: R1 = [.r,a.[ ;
 a is Real by XREAL_0:def_1;
then  upper_bound [.r,a.[ in  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  by A22, A28, A36;
hence  not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty ; ::_thesis: verum
end;
suppose ex a being real number st
( r <= a & a < s & R1 = ].a,s.] ) ; ::_thesis: not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
hence  not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty by A27, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & R1 = ].a,b.[ ) ; ::_thesis: not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty
hence  not  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is empty by A27, XXREAL_1:4; ::_thesis: verum
end;
end;
end;
A37: G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  is finite by A8;
  {  (lower_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is finite from FRAENKEL:sch_21(A37);
then reconsider RM =  {  (lower_bound ].c,s.]) where c is Real : ].c,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  as non empty finite Subset of REAL by A15, A33, A16;
 F c= bool REAL
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in F or a in bool REAL )
assume  a in F ; ::_thesis: a in bool REAL
then  a c= REAL by A4, XBOOLE_1:1;
hence  a in bool REAL ; ::_thesis: verum
end;
then  G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  c= bool REAL by A11, XBOOLE_1:1;
then reconsider X = C as non empty finite Subset-Family of REAL by A8, A22, A28, XBOOLE_1:1;
  {  (upper_bound E) where E is Subset of (Closed-Interval-TSpace (r,s)) : E in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  is finite from FRAENKEL:sch_21(A37);
then reconsider LM =  {  (upper_bound [.r,b.[) where b is Real : [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }   }  as non empty finite Subset of REAL by A13, A35, A14;
reconsider kL = max LM as Real by XREAL_0:def_1;
set LEWY = [.r,kL.[;
 kL in LM by XXREAL_2:def_8;
then consider b being Real such that
A38: kL = upper_bound [.r,b.[ and
A39: [.r,b.[ in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ;
A40: union G = [#] (Closed-Interval-TSpace (r,s)) by A7, SETFAM_1:45;
A41: now__::_thesis:_not_{}_in_G_\__{__X_where_X_is_Subset_of_(Closed-Interval-TSpace_(r,s))_:_(_X_in_G_&_ex_Y_being_Subset_of_(Closed-Interval-TSpace_(r,s))_st_
(_Y_in_G_&_X_c<_Y_)_)__}_
consider x being set  such that
A42: x in the carrier of (Closed-Interval-TSpace (r,s)) by XBOOLE_0:def_1;
consider g being set  such that
A43: x in g and
A44: g in G by A40, A42, TARSKI:def_4;
 {} c= g by XBOOLE_1:2;
then A45: {} c< g by A43, XBOOLE_0:def_8;
assume A46: {} in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ; ::_thesis: contradiction
then  {} in G by XBOOLE_0:def_5;
then  {} in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A44, A45;
hence  contradiction by A46, XBOOLE_0:def_5; ::_thesis: verum
end;
then A47: upper_bound [.r,kL.[ = kL by A38, A39, Th5, XXREAL_1:27;
A48: r < b by A41, A39, XXREAL_1:27;
then  r < kL by A38, Th5;
then A49: lower_bound [.r,kL.[ = r by Th4;
reconsider LEWY = [.r,kL.[ as non empty Subset of (Closed-Interval-TSpace (r,s)) by A41, A38, A39, Th5, XXREAL_1:27;
A50: kL = b by A41, A38, A39, Th5, XXREAL_1:27;
A51: for A being Subset of (Closed-Interval-TSpace (r,s)) st r in A & A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  holds
A = LEWY
proof
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( r in A & A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  implies A = LEWY )
assume that
A52: r in A and
A53: A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ; ::_thesis: A = LEWY
A54: ( A in F & A is open ) by B2, A11, A53, TOPS_2:def_1;
A55: now__::_thesis:_(_(_for_a_being_real_number_holds_
(_not_r_<=_a_or_not_a_<_s_or_not_A_=_].a,s.]_)_)_&_(_for_a,_b_being_real_number_holds_
(_not_r_<=_a_or_not_a_<_b_or_not_b_<=_s_or_not_A_=_].a,b.[_)_)_)
assume A56: ( ex a being real number st
( r <= a & a < s & A = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ) ; ::_thesis: contradiction
percases ( ex a being real number st
( r <= a & a < s & A = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ) by A56;
suppose ex a being real number st
( r <= a & a < s & A = ].a,s.] ) ; ::_thesis: contradiction
hence  contradiction by A52, XXREAL_1:2; ::_thesis: verum
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ; ::_thesis: contradiction
hence  contradiction by A52, XXREAL_1:4; ::_thesis: verum
end;
end;
end;
 A is connected by B3, A11, A53, Def1;
then consider ak being real number  such that
A57: r < ak and
 ak <= s and
A58: A = [.r,ak.[ by B4, A3, A52, A54, A55, Th44;
A59: ak is Real by XREAL_0:def_1;
A60: A c= LEWY
proof
 upper_bound A = ak by A57, A58, Th5;
then  ak in LM by A53, A58, A59;
then A61: ak <= kL by XXREAL_2:def_8;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A or a in LEWY )
assume A62: a in A ; ::_thesis: a in LEWY
then  a in [.r,s.] by A4;
then reconsider a = a as Real ;
 a < ak by A58, A62, XXREAL_1:3;
then A63: a < kL by A61, XXREAL_0:2;
 r <= a by A58, A62, XXREAL_1:3;
hence  a in LEWY by A63, XXREAL_1:3; ::_thesis: verum
end;
assume  A <> LEWY ; ::_thesis: contradiction
then  A c< LEWY by A60, XBOOLE_0:def_8;
then  A in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A39, A50, A53;
hence  contradiction by A53, XBOOLE_0:def_5; ::_thesis: verum
end;
then reconsider LLEWY = LEWY as Element of X by A22, A27, A28;
reconsider pP = min RM as Real by XREAL_0:def_1;
set PRAWY = ].pP,s.];
 pP in RM by XXREAL_2:def_7;
then consider b being Real such that
A64: pP = lower_bound ].b,s.] and
A65: ].b,s.] in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ;
A66: lower_bound ].pP,s.] = pP by A41, A64, A65, Th6, XXREAL_1:26;
A67: b < s by A41, A65, XXREAL_1:26;
then  pP < s by A64, Th6;
then A68: upper_bound ].pP,s.] = s by Th7;
reconsider PRAWY = ].pP,s.] as non empty Subset of (Closed-Interval-TSpace (r,s)) by A41, A64, A65, Th6, XXREAL_1:26;
A69: pP = b by A41, A64, A65, Th6, XXREAL_1:26;
A70: for A being Subset of (Closed-Interval-TSpace (r,s)) st A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & A <> LEWY & A <> PRAWY holds
ex a, b being Real st
( a < b & A = ].a,b.[ )
proof
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & A <> LEWY & A <> PRAWY implies ex a, b being Real st
( a < b & A = ].a,b.[ ) )
assume that
A71: A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  and
A72: A <> LEWY and
A73: A <> PRAWY ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
A74: ( A in F & A is open & A is connected ) by B3, B2, A11, A71, Def1, TOPS_2:def_1;
percases ( ex a being real number st
( r < a & a <= s & A = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & A = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ) by B4, A3, A41, A71, A74, Th44;
suppose ex a being real number st
( r < a & a <= s & A = [.r,a.[ ) ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then consider a being real number  such that
 r < a and
 a <= s and
A75: A = [.r,a.[ ;
percases ( a <= kL or a > kL ) ;
suppose a <= kL ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then  A c= LEWY by A75, XXREAL_1:38;
then  A c< LEWY by A72, XBOOLE_0:def_8;
then  A in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A39, A50, A71;
hence  ex a, b being Real st
( a < b & A = ].a,b.[ ) by A71, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose a > kL ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then  LEWY c= A by A75, XXREAL_1:38;
then  LEWY c< A by A72, XBOOLE_0:def_8;
then  LEWY in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A39, A50, A71;
hence  ex a, b being Real st
( a < b & A = ].a,b.[ ) by A39, A50, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
suppose ex a being real number st
( r <= a & a < s & A = ].a,s.] ) ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then consider a being real number  such that
 r <= a and
 a < s and
A76: A = ].a,s.] ;
percases ( a >= pP or a < pP ) ;
suppose a >= pP ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then  A c= PRAWY by A76, XXREAL_1:42;
then  A c< PRAWY by A73, XBOOLE_0:def_8;
then  A in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A65, A69, A71;
hence  ex a, b being Real st
( a < b & A = ].a,b.[ ) by A71, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose a < pP ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then  PRAWY c= A by A76, XXREAL_1:42;
then  PRAWY c< A by A73, XBOOLE_0:def_8;
then  PRAWY in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A65, A69, A71;
hence  ex a, b being Real st
( a < b & A = ].a,b.[ ) by A65, A69, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ; ::_thesis: ex a, b being Real st
( a < b & A = ].a,b.[ )
then consider a, b being real number  such that
 r <= a and
A77: a < b and
 b <= s and
A78: A = ].a,b.[ ;
reconsider a = a, b = b as Real by XREAL_0:def_1;
take  a ; ::_thesis: ex b being Real st
( a < b & A = ].a,b.[ )
take  b ; ::_thesis: ( a < b & A = ].a,b.[ )
thus  ( a < b & A = ].a,b.[ ) by A77, A78; ::_thesis: verum
end;
end;
end;
A79: for A being Subset of (Closed-Interval-TSpace (r,s)) st A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & upper_bound A in A holds
A = PRAWY
proof
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & upper_bound A in A implies A = PRAWY )
assume that
A80: A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  and
A81: upper_bound A in A and
A82: A <> PRAWY ; ::_thesis: contradiction
 A <> LEWY by A47, A81, XXREAL_1:3;
then consider a, b being Real such that
A83: a < b and
A84: A = ].a,b.[ by A70, A80, A82;
 upper_bound A = b by A83, A84, TOPREAL6:17;
hence  contradiction by A81, A84, XXREAL_1:4; ::_thesis: verum
end;
defpred S2[ set , set , set ] means  ex S being Element of X st
( S = $2 & upper_bound S in $3 );
A85: X c= F by A11, A22, XBOOLE_1:1;
A86: for Z being Subset of REAL st Z in X holds
Z is non empty open connected Subset of T
proof
let Z be Subset of REAL; ::_thesis: ( Z in X implies Z is non empty open connected Subset of T )
assume A87: Z in X ; ::_thesis: Z is non empty open connected Subset of T
then  ( not Z is empty & Z is interval ) by A41, A12, A22;
hence  Z is non empty open connected Subset of T by B2, A85, A87, Th43, TOPS_2:def_1; ::_thesis: verum
end;
A88: for n being Element of NAT st 1 <= n & n < card X holds
for x being Element of X ex y being Element of X st S2[n,x,y]
proof
let n be Element of NAT ; ::_thesis: ( 1 <= n & n < card X implies for x being Element of X ex y being Element of X st S2[n,x,y] )
assume that
 1 <= n and
 n < card X ; ::_thesis: for x being Element of X ex y being Element of X st S2[n,x,y]
let x be Element of X; ::_thesis: ex y being Element of X st S2[n,x,y]
reconsider x1 = x as Subset of REAL ;
A89: not x1 is empty by A86;
A90: x c= union X by ZFMISC_1:74;
then  x c= [.r,s.] by A4, A23;
then  x1 is bounded_above by XXREAL_2:43;
then  upper_bound x is Element of (Closed-Interval-TSpace (r,s)) by A4, A23, A89, A90, Th2;
then consider y being set  such that
A91: upper_bound x in y and
A92: y in X by A23, TARSKI:def_4;
reconsider y = y as Element of X by A92;
take  y ; ::_thesis: S2[n,x,y]
take  x ; ::_thesis: ( x = x & upper_bound x in y )
thus  ( x = x & upper_bound x in y ) by A91; ::_thesis: verum
end;
consider IT being FinSequence of X such that
A93: len IT = card X and
A94: ( IT . 1 = LLEWY or card X = 0 ) and
A95: for n being Element of NAT st 1 <= n & n < card X holds
S2[n,IT . n,IT . (n + 1)] from RECDEF_1:sch_4(A88);
A96: rng IT c= X ;
 rng IT c= bool REAL by XBOOLE_1:1;
then reconsider IT = IT as FinSequence of bool REAL by FINSEQ_1:def_4;
A97: not IT is empty by A93;
then A98: not rng IT is empty ;
then A99: 1 in dom IT by FINSEQ_3:32;
then A100: IT /. 1 = IT . 1 by PARTFUN1:def_6;
A101: for n being Nat st n in dom IT holds
( IT . n in X & IT /. n in X )
proof
let n be Nat; ::_thesis: ( n in dom IT implies ( IT . n in X & IT /. n in X ) )
assume  n in dom IT ; ::_thesis: ( IT . n in X & IT /. n in X )
then  ( IT . n = IT /. n & IT . n in rng IT ) by FUNCT_1:def_3, PARTFUN1:def_6;
hence  ( IT . n in X & IT /. n in X ) by A96; ::_thesis: verum
end;
A102: for n being Nat st n in dom IT holds
( IT . n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )
proof
let n be Nat; ::_thesis: ( n in dom IT implies ( IT . n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) )
assume  n in dom IT ; ::_thesis: ( IT . n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )
then  ( IT . n = IT /. n & IT . n in X ) by A101, PARTFUN1:def_6;
hence  ( IT . n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) by A22; ::_thesis: verum
end;
A103: for i being Nat st i in dom IT holds
for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. i) holds
ex j being Nat st
( j in dom IT & j <= i & x in IT /. j )
proof
defpred S3[ Nat] means ( $1 in dom IT implies for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. $1) holds
ex j being Nat st
( j in dom IT & j <= $1 & x in IT /. j ) );
A104: for n being Nat st S3[n] holds
S3[n + 1]
proof
let n be Nat; ::_thesis: ( S3[n] implies S3[n + 1] )
assume that
A105: S3[n] and
A106: n + 1 in dom IT ; ::_thesis: for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. (n + 1)) holds
ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
A107: IT /. (n + 1) = IT . (n + 1) by A106, PARTFUN1:def_6;
let x be Point of (Closed-Interval-TSpace (r,s)); ::_thesis: ( x < upper_bound (IT /. (n + 1)) implies ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j ) )
assume A108: x < upper_bound (IT /. (n + 1)) ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
percases ( n = 0 or n in dom IT ) by A106, TOPREALA:2;
supposeA109: n = 0 ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
take  1 ; ::_thesis: ( 1 in dom IT & 1 <= n + 1 & x in IT /. 1 )
thus  1 in dom IT by A98, FINSEQ_3:32; ::_thesis: ( 1 <= n + 1 & x in IT /. 1 )
thus  1 <= n + 1 by A109; ::_thesis: x in IT /. 1
 r <= x by A4, XXREAL_1:1;
hence  x in IT /. 1 by A47, A94, A108, A107, A109, XXREAL_1:3; ::_thesis: verum
end;
supposeA110: n in dom IT ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
 n + 1 <= len IT by A106, FINSEQ_3:25;
then A111: n < len IT by NAT_1:13;
 1 <= n by A110, FINSEQ_3:25;
then A112: ex S being Element of X st
( S = IT . n & upper_bound S in IT . (n + 1) ) by A93, A95, A110, A111;
 IT /. (n + 1) in X by A101, A106;
then A113: IT /. (n + 1) is bounded_below by A5;
 IT /. n = IT . n by A110, PARTFUN1:def_6;
then A114: lower_bound (IT /. (n + 1)) <= upper_bound (IT /. n) by A107, A113, A112, SEQ_4:def_2;
A115: ( IT /. (n + 1) is interval Subset of REAL & not IT /. (n + 1) is empty ) by A41, A12, A102, A106;
percases ( x < upper_bound (IT /. n) or x = upper_bound (IT /. n) or x > upper_bound (IT /. n) ) by XXREAL_0:1;
suppose x < upper_bound (IT /. n) ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
then consider j being Nat such that
A116: j in dom IT and
A117: j <= n and
A118: x in IT /. j by A105, A110;
take  j ; ::_thesis: ( j in dom IT & j <= n + 1 & x in IT /. j )
thus  j in dom IT by A116; ::_thesis: ( j <= n + 1 & x in IT /. j )
 j + 0 < n + 1 by A117, XREAL_1:8;
hence  j <= n + 1 ; ::_thesis: x in IT /. j
thus  x in IT /. j by A118; ::_thesis: verum
end;
supposeA119: x = upper_bound (IT /. n) ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
take  n + 1 ; ::_thesis: ( n + 1 in dom IT & n + 1 <= n + 1 & x in IT /. (n + 1) )
thus  n + 1 in dom IT by A106; ::_thesis: ( n + 1 <= n + 1 & x in IT /. (n + 1) )
thus  n + 1 <= n + 1 ; ::_thesis: x in IT /. (n + 1)
thus  x in IT /. (n + 1) by A107, A110, A112, A119, PARTFUN1:def_6; ::_thesis: verum
end;
supposeA120: x > upper_bound (IT /. n) ; ::_thesis: ex j being Nat st
( j in dom IT & j <= n + 1 & x in IT /. j )
take  n + 1 ; ::_thesis: ( n + 1 in dom IT & n + 1 <= n + 1 & x in IT /. (n + 1) )
thus  n + 1 in dom IT by A106; ::_thesis: ( n + 1 <= n + 1 & x in IT /. (n + 1) )
thus  n + 1 <= n + 1 ; ::_thesis: x in IT /. (n + 1)
 lower_bound (IT /. (n + 1)) < x by A114, A120, XXREAL_0:2;
hence  x in IT /. (n + 1) by A108, A115, Th30; ::_thesis: verum
end;
end;
end;
end;
end;
A121: S3[ 0 ] by FINSEQ_3:24;
A122: for n being Nat holds S3[n] from NAT_1:sch_2(A121, A104);
let i be Nat; ::_thesis: ( i in dom IT implies for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. i) holds
ex j being Nat st
( j in dom IT & j <= i & x in IT /. j ) )
assume  i in dom IT ; ::_thesis: for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. i) holds
ex j being Nat st
( j in dom IT & j <= i & x in IT /. j )
hence  for x being Point of (Closed-Interval-TSpace (r,s)) st x < upper_bound (IT /. i) holds
ex j being Nat st
( j in dom IT & j <= i & x in IT /. j ) by A122; ::_thesis: verum
end;
A123: s in ].b,s.] by A67, XXREAL_1:2;
A124: for i being Nat st i in dom IT holds
for j being Nat st j in dom IT & i < j holds
ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. j & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
proof
let i be Nat; ::_thesis: ( i in dom IT implies for j being Nat st j in dom IT & i < j holds
ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. j & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) )
assume A125: i in dom IT ; ::_thesis: for j being Nat st j in dom IT & i < j holds
ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. j & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
defpred S3[ Nat] means ( $1 in dom IT & i < $1 implies ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. $1 & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) );
A126: for n being Nat st S3[n] holds
S3[n + 1]
proof
let n be Nat; ::_thesis: ( S3[n] implies S3[n + 1] )
assume that
A127: S3[n] and
A128: n + 1 in dom IT ; ::_thesis: ( not i < n + 1 or ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) )
A129: IT /. (n + 1) = IT . (n + 1) by A128, PARTFUN1:def_6;
assume A130: i < n + 1 ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
then A131: i <= n by NAT_1:13;
percases ( n = 0 or n in dom IT ) by A128, TOPREALA:2;
suppose n = 0 ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
then  i = 0 by A130, NAT_1:13;
hence  ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) by A125, FINSEQ_3:24; ::_thesis: verum
end;
supposeA132: n in dom IT ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
then A133: IT /. n in X by A101;
then A134: IT /. n is bounded_above by A5;
A135: IT /. n = IT . n by A132, PARTFUN1:def_6;
then  IT /. n in rng IT by A132, FUNCT_1:def_3;
then A136: ( not IT /. n is empty & IT /. n is Subset of (Closed-Interval-TSpace (r,s)) ) by A86, A96;
then  upper_bound (IT /. n) in [.r,s.] by A4, A134, Th2;
then A137: upper_bound (IT /. n) <= s by XXREAL_1:1;
A138: IT /. (n + 1) in X by A101, A128;
A139: 1 <= n by A132, FINSEQ_3:25;
A140: IT /. (n + 1) in rng IT by A128, A129, FUNCT_1:def_3;
then A141: IT /. (n + 1) is open connected Subset of (Closed-Interval-TSpace (r,s)) by A86, A96;
then A142: IT /. (n + 1) is interval Subset of REAL by Th43;
A143: n + 1 <= len IT by A128, FINSEQ_3:25;
then  ( n is Element of NAT & n < card X ) by A93, NAT_1:13, ORDINAL1:def_12;
then consider S being Element of X such that
A144: S = IT . n and
A145: upper_bound S in IT . (n + 1) by A95, A139;
 IT /. (n + 1) is bounded_below by A5, A141;
then A146: lower_bound (IT /. (n + 1)) <= upper_bound S by A129, A145, SEQ_4:def_2;
A147: IT /. (n + 1) is bounded_above by A5, A141;
then A148: upper_bound S <= upper_bound (IT /. (n + 1)) by A129, A145, SEQ_4:def_1;
A149: not IT /. (n + 1) is empty by A86, A96, A140;
then  upper_bound (IT /. (n + 1)) in [.r,s.] by A4, A141, A147, Th2;
then A150: upper_bound (IT /. (n + 1)) <= s by XXREAL_1:1;
percases ( i < n or i = n ) by A131, XXREAL_0:1;
suppose i < n ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
then consider y being Point of (Closed-Interval-TSpace (r,s)) such that
A151: y in IT /. n and
A152: for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y by A127, A132;
A153: y <= upper_bound (IT /. n) by A134, A151, SEQ_4:def_1;
percases ( upper_bound S < upper_bound (IT /. (n + 1)) or upper_bound S = upper_bound (IT /. (n + 1)) ) by A148, XXREAL_0:1;
supposeA154: upper_bound S < upper_bound (IT /. (n + 1)) ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
set y1 = ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2;
A155: upper_bound S < ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2 by A154, XREAL_1:226;
 upper_bound S in [.r,s.] by A4, A135, A134, A136, A144, Th2;
then  r <= upper_bound S by XXREAL_1:1;
then A156: r <= ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2 by A155, XXREAL_0:2;
 ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2 < upper_bound (IT /. (n + 1)) by A154, XREAL_1:226;
then  ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2 < s by A150, XXREAL_0:2;
then reconsider y1 = ((upper_bound S) + (upper_bound (IT /. (n + 1)))) / 2 as Point of (Closed-Interval-TSpace (r,s)) by A4, A156, XXREAL_1:1;
take  y1 ; ::_thesis: ( y1 in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1 ) )
 lower_bound (IT /. (n + 1)) < y1 by A146, A155, XXREAL_0:2;
hence  y1 in IT /. (n + 1) by A142, A149, A154, Th30, XREAL_1:226; ::_thesis: for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1
let x be Point of (Closed-Interval-TSpace (r,s)); ::_thesis: ( x in IT /. i implies x < y1 )
assume  x in IT /. i ; ::_thesis: x < y1
then  x < upper_bound (IT /. n) by A152, A153, XXREAL_0:2;
hence  x < y1 by A135, A144, A155, XXREAL_0:2; ::_thesis: verum
end;
supposeA157: upper_bound S = upper_bound (IT /. (n + 1)) ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
reconsider y1 = s as Point of (Closed-Interval-TSpace (r,s)) by B4, A4, XXREAL_1:1;
take  y1 ; ::_thesis: ( y1 in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1 ) )
 IT /. (n + 1) = PRAWY by A22, A79, A129, A145, A138, A157;
hence  y1 in IT /. (n + 1) by A67, A69, XXREAL_1:2; ::_thesis: for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1
let x be Point of (Closed-Interval-TSpace (r,s)); ::_thesis: ( x in IT /. i implies x < y1 )
assume  x in IT /. i ; ::_thesis: x < y1
then  x < upper_bound (IT /. n) by A152, A153, XXREAL_0:2;
hence  x < y1 by A137, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
supposeA158: i = n ; ::_thesis: ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) )
reconsider y1 = upper_bound (IT /. n) as Element of (Closed-Interval-TSpace (r,s)) by A4, A134, A136, Th2;
take  y1 ; ::_thesis: ( y1 in IT /. (n + 1) & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1 ) )
thus  y1 in IT /. (n + 1) by A129, A132, A144, A145, PARTFUN1:def_6; ::_thesis: for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y1
let x be Point of (Closed-Interval-TSpace (r,s)); ::_thesis: ( x in IT /. i implies x < y1 )
assume A159: x in IT /. i ; ::_thesis: x < y1
A160: now__::_thesis:_not_x_=_upper_bound_(IT_/._n)
set IT1 = IT | (Seg n);
A161: rng (IT | (Seg n)) c= rng IT by RELAT_1:70;
 rng (IT | (Seg n)) c= bool the carrier of (Closed-Interval-TSpace (r,s))
proof
let A be set ; :: according to TARSKI:def_3 ::_thesis: ( not A in rng (IT | (Seg n)) or A in bool the carrier of (Closed-Interval-TSpace (r,s)) )
assume  A in rng (IT | (Seg n)) ; ::_thesis: A in bool the carrier of (Closed-Interval-TSpace (r,s))
then  A in rng IT by A161;
then  A in X by A96;
hence  A in bool the carrier of (Closed-Interval-TSpace (r,s)) ; ::_thesis: verum
end;
then reconsider FI = rng (IT | (Seg n)) as Subset-Family of (Closed-Interval-TSpace (r,s)) ;
assume  x = upper_bound (IT /. n) ; ::_thesis: contradiction
then A162: IT /. n = PRAWY by A22, A79, A133, A158, A159;
A163: now__::_thesis:_not_FI_<>_X
 union FI = the carrier of (Closed-Interval-TSpace (r,s))
proof
thus  union FI c= the carrier of (Closed-Interval-TSpace (r,s)) ; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of (Closed-Interval-TSpace (r,s)) c= union FI
let l be set ; :: according to TARSKI:def_3 ::_thesis: ( not l in the carrier of (Closed-Interval-TSpace (r,s)) or l in union FI )
assume  l in the carrier of (Closed-Interval-TSpace (r,s)) ; ::_thesis: l in union FI
then reconsider l = l as Point of (Closed-Interval-TSpace (r,s)) ;
percases ( l < upper_bound (IT /. n) or l >= upper_bound (IT /. n) ) ;
suppose l < upper_bound (IT /. n) ; ::_thesis: l in union FI
then consider j being Nat such that
A164: j in dom IT and
A165: j <= n and
A166: l in IT /. j by A103, A132;
 1 <= j by A164, FINSEQ_3:25;
then  j in Seg n by A165, FINSEQ_1:1;
then A167: IT . j in FI by A164, FUNCT_1:50;
 IT . j = IT /. j by A164, PARTFUN1:def_6;
hence  l in union FI by A166, A167, TARSKI:def_4; ::_thesis: verum
end;
supposeA168: l >= upper_bound (IT /. n) ; ::_thesis: l in union FI
 n in Seg n by A139, FINSEQ_1:1;
then A169: IT . n in FI by A132, FUNCT_1:50;
 l <= s by A4, XXREAL_1:1;
then  l = s by A68, A162, A168, XXREAL_0:1;
hence  l in union FI by A69, A123, A135, A162, A169, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
then A170: FI is Cover of (Closed-Interval-TSpace (r,s)) by SETFAM_1:def_11;
assume A171: FI <> X ; ::_thesis: contradiction
A172: FI c= X by A96, A161, XBOOLE_1:1;
then  FI c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A22, XBOOLE_1:1;
then A173: FI in  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by A170;
then  [FI,M] in RelIncl  {  C where C is Subset-Family of (Closed-Interval-TSpace (r,s)) : ( C is Cover of (Closed-Interval-TSpace (r,s)) & C c= G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  )  }  by A17, A19, A20, A172, WELLORD2:def_1;
hence  contradiction by A17, A18, A20, A171, A173, ORDERS_1:def_12; ::_thesis: verum
end;
 Seg n c= dom IT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Seg n or x in dom IT )
A174: n + 0 <= n + 1 by XREAL_1:6;
assume A175: x in Seg n ; ::_thesis: x in dom IT
then reconsider x = x as Nat ;
 x <= n by A175, FINSEQ_1:1;
then  x <= n + 1 by A174, XXREAL_0:2;
then A176: x <= len IT by A143, XXREAL_0:2;
 1 <= x by A175, FINSEQ_1:1;
hence  x in dom IT by A176, FINSEQ_3:25; ::_thesis: verum
end;
then  dom (IT | (Seg n)) = Seg n by RELAT_1:62;
then  ( card (rng (IT | (Seg n))) <= card (dom (IT | (Seg n))) & card (dom (IT | (Seg n))) = n ) by CARD_2:47, FINSEQ_1:57;
then  n + 1 <= n + 0 by A93, A143, A163, XXREAL_0:2;
hence  contradiction by XREAL_1:6; ::_thesis: verum
end;
 x <= upper_bound (IT /. n) by A134, A158, A159, SEQ_4:def_1;
hence  x < y1 by A160, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
end;
A177: S3[ 0 ] ;
 for n being Nat holds S3[n] from NAT_1:sch_2(A177, A126);
hence  for j being Nat st j in dom IT & i < j holds
ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. j & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) ; ::_thesis: verum
end;
A178: IT is one-to-one
proof
let i, j be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not i in dom IT or not j in dom IT or not IT . i = IT . j or i = j )
assume that
A179: ( i in dom IT & j in dom IT ) and
A180: IT . i = IT . j ; ::_thesis: i = j
A181: ( IT /. i = IT . i & IT /. j = IT . j ) by A179, PARTFUN1:def_6;
assume A182: i <> j ; ::_thesis: contradiction
reconsider i = i, j = j as Nat by A179;
percases ( i < j or j < i ) by A182, XXREAL_0:1;
suppose i < j ; ::_thesis: contradiction
then  ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. j & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. i holds
x < y ) ) by A124, A179;
hence  contradiction by A180, A181; ::_thesis: verum
end;
suppose j < i ; ::_thesis: contradiction
then  ex y being Point of (Closed-Interval-TSpace (r,s)) st
( y in IT /. i & ( for x being Point of (Closed-Interval-TSpace (r,s)) st x in IT /. j holds
x < y ) ) by A124, A179;
hence  contradiction by A180, A181; ::_thesis: verum
end;
end;
end;
A183: for i, j being Nat st i in dom IT & j in dom IT & i <> j holds
IT /. i <> IT /. j
proof
let i, j be Nat; ::_thesis: ( i in dom IT & j in dom IT & i <> j implies IT /. i <> IT /. j )
assume that
A184: ( i in dom IT & j in dom IT ) and
A185: i <> j ; ::_thesis: IT /. i <> IT /. j
 ( IT /. i = IT . i & IT /. j = IT . j ) by A184, PARTFUN1:def_6;
hence  IT /. i <> IT /. j by A178, A184, A185, FUNCT_1:def_4; ::_thesis: verum
end;
A186: for A being Subset of (Closed-Interval-TSpace (r,s)) st s in A & A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  holds
A = PRAWY
proof
let A be Subset of (Closed-Interval-TSpace (r,s)); ::_thesis: ( s in A & A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  implies A = PRAWY )
assume that
A187: s in A and
A188: A in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ; ::_thesis: A = PRAWY
A189: ( A in F & A is open ) by B2, A11, A188, TOPS_2:def_1;
A190: now__::_thesis:_(_(_for_a_being_real_number_holds_
(_not_r_<_a_or_not_a_<=_s_or_not_A_=_[.r,a.[_)_)_&_(_for_a,_b_being_real_number_holds_
(_not_r_<=_a_or_not_a_<_b_or_not_b_<=_s_or_not_A_=_].a,b.[_)_)_)
assume A191: ( ex a being real number st
( r < a & a <= s & A = [.r,a.[ ) or ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ) ; ::_thesis: contradiction
percases ( ex a being real number st
( r < a & a <= s & A = [.r,a.[ ) or ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ) by A191;
suppose ex a being real number st
( r < a & a <= s & A = [.r,a.[ ) ; ::_thesis: contradiction
hence  contradiction by A187, XXREAL_1:3; ::_thesis: verum
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & A = ].a,b.[ ) ; ::_thesis: contradiction
hence  contradiction by A187, XXREAL_1:4; ::_thesis: verum
end;
end;
end;
 A is connected by B3, A11, A188, Def1;
then consider ak being real number  such that
 r <= ak and
A192: ak < s and
A193: A = ].ak,s.] by B4, A3, A187, A189, A190, Th44;
A194: ak is Real by XREAL_0:def_1;
A195: A c= PRAWY
proof
 lower_bound A = ak by A192, A193, Th6;
then  ak in RM by A188, A193, A194;
then A196: pP <= ak by XXREAL_2:def_7;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A or a in PRAWY )
assume A197: a in A ; ::_thesis: a in PRAWY
then  a in [.r,s.] by A4;
then reconsider a = a as Real ;
 ak < a by A193, A197, XXREAL_1:2;
then A198: pP < a by A196, XXREAL_0:2;
 a <= s by A193, A197, XXREAL_1:2;
hence  a in PRAWY by A198, XXREAL_1:2; ::_thesis: verum
end;
assume  A <> PRAWY ; ::_thesis: contradiction
then  A c< PRAWY by A195, XBOOLE_0:def_8;
then  A in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A65, A69, A188;
hence  contradiction by A188, XBOOLE_0:def_5; ::_thesis: verum
end;
take  IT ; ::_thesis: ( rng IT c= F & union (rng IT) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) & ( n + 2 <= len IT implies upper_bound (IT /. n) <= lower_bound (IT /. (n + 2)) ) ) ) & ( [.r,s.] in F implies IT = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) ) ) )
thus  rng IT c= F by A85, A96, XBOOLE_1:1; ::_thesis: ( union (rng IT) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) & ( n + 2 <= len IT implies upper_bound (IT /. n) <= lower_bound (IT /. (n + 2)) ) ) ) & ( [.r,s.] in F implies IT = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) ) ) )
 dom IT = Seg (len IT) by FINSEQ_1:def_3;
then A199: card (dom IT) = card X by A93, FINSEQ_1:57;
 IT is Function of (dom IT),X by A96, FUNCT_2:2;
then A200: rng IT = X by A199, A178, FINSEQ_4:63;
hence  union (rng IT) = [.r,s.] by A4, A23; ::_thesis: ( ( for n being Nat st 1 <= n holds
( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) & ( n + 2 <= len IT implies upper_bound (IT /. n) <= lower_bound (IT /. (n + 2)) ) ) ) & ( [.r,s.] in F implies IT = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) ) ) )
 ex Z being set st
( s in Z & Z in C ) by A24, A4, A23, TARSKI:def_4;
then  PRAWY in X by A22, A186;
then consider i being set  such that
A201: i in dom IT and
A202: IT . i = PRAWY by A200, FUNCT_1:def_3;
reconsider i = i as Element of NAT by A201;
A203: i <= len IT by A201, FINSEQ_3:25;
A204: IT /. i = IT . i by A201, PARTFUN1:def_6;
A205: 1 <= i by A201, FINSEQ_3:25;
A206: now__::_thesis:_not_i_<>_len_IT
assume  i <> len IT ; ::_thesis: contradiction
then A207: i < len IT by A203, XXREAL_0:1;
then A208: ex S being Element of X st
( S = IT . i & upper_bound S in IT . (i + 1) ) by A93, A95, A205;
 ( 0 + 1 <= i + 1 & i + 1 <= len IT ) by A207, NAT_1:13;
then A209: i + 1 in dom IT by FINSEQ_3:25;
then  ( IT /. (i + 1) = IT . (i + 1) & IT /. (i + 1) in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  ) by A102, PARTFUN1:def_6;
then  i + 0 = i + 1 by A68, A186, A183, A201, A202, A204, A208, A209;
hence  contradiction ; ::_thesis: verum
end;
A210: len IT in dom IT by A97, FINSEQ_5:6;
A211: for n being Nat st 1 < n & n < len IT holds
ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
proof
let n be Nat; ::_thesis: ( 1 < n & n < len IT implies ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) )
assume that
A212: 1 < n and
A213: n < len IT ; ::_thesis: ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
A214: n in dom IT by A212, A213, FINSEQ_3:25;
then  IT . n in rng IT by FUNCT_1:def_3;
then A215: IT /. n in rng IT by A214, PARTFUN1:def_6;
then A216: IT /. n in X by A96;
then A217: ( IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  & IT /. n in F ) by A22, A85;
A218: IT /. n is open connected Subset of (Closed-Interval-TSpace (r,s)) by A86, A96, A215;
percases ( ex a being real number st
( r < a & a <= s & IT /. n = [.r,a.[ ) or ex a being real number st
( r <= a & a < s & IT /. n = ].a,s.] ) or ex a, b being real number st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) ) by B4, A3, A41, A217, A218, Th44;
suppose ex a being real number st
( r < a & a <= s & IT /. n = [.r,a.[ ) ; ::_thesis: ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
then consider a being real number  such that
A219: r < a and
 a <= s and
A220: IT /. n = [.r,a.[ ;
 r in [.r,a.[ by A219, XXREAL_1:3;
hence  ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) by A22, A51, A94, A99, A100, A183, A212, A214, A216, A220; ::_thesis: verum
end;
suppose ex a being real number st
( r <= a & a < s & IT /. n = ].a,s.] ) ; ::_thesis: ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
then consider a being real number  such that
 r <= a and
A221: a < s and
A222: IT /. n = ].a,s.] ;
 ( upper_bound ].a,s.] = s & s in ].a,s.] ) by A221, Th7, XXREAL_1:2;
hence  ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) by A22, A79, A210, A183, A202, A204, A206, A213, A214, A216, A222; ::_thesis: verum
end;
suppose ex a, b being real number st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) ; ::_thesis: ex a, b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
then consider a, b being real number  such that
A223: ( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) ;
reconsider a = a, b = b as Real by XREAL_0:def_1;
take  a ; ::_thesis: ex b being Real st
( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
take  b ; ::_thesis: ( r <= a & a < b & b <= s & IT /. n = ].a,b.[ )
thus  ( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) by A223; ::_thesis: verum
end;
end;
end;
A224: now__::_thesis:_for_n_being_Nat_st_1_<=_n_holds_
(_(_n_<=_len_IT_implies_not_IT_/._n_is_empty_)_&_(_n_+_1_<=_len_IT_implies_(_not_lower_bound_(IT_/._n)_>_lower_bound_(IT_/._(n_+_1))_&_upper_bound_(IT_/._n)_<=_upper_bound_(IT_/._(n_+_1))_&_lower_bound_(IT_/._(n_+_1))_<_upper_bound_(IT_/._n)_)_)_)
let n be Nat; ::_thesis: ( 1 <= n implies ( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( not lower_bound (IT /. n) > lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) ) )
assume A225: 1 <= n ; ::_thesis: ( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( not lower_bound (IT /. n) > lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) )
reconsider m = n as Element of NAT by ORDINAL1:def_12;
hereby  ::_thesis: ( n + 1 <= len IT implies ( not lower_bound (IT /. n) > lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) )
assume  n <= len IT ; ::_thesis: not IT /. n is empty
then  ( m in dom IT & IT /. n = IT . n ) by A225, FINSEQ_3:25, FINSEQ_4:15;
then  IT /. n in rng IT by FUNCT_1:def_3;
then  IT /. n in X by A96;
hence  not IT /. n is empty by A41, A22; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume A226: n + 1 <= len IT ; ::_thesis: ( not lower_bound (IT /. n) > lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) )
then A227: m < len IT by NAT_1:13;
then A228: IT /. n = IT . n by A225, FINSEQ_4:15;
A229: m in dom IT by A225, A227, FINSEQ_3:25;
then  IT /. n in rng IT by A228, FUNCT_1:def_3;
then A230: IT /. n in X by A96;
then A231: IT /. n in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A22;
A232: IT /. n is non empty real-bounded interval Subset of REAL by A5, A41, A12, A22, A230;
A233: ex S being Element of X st
( S = IT . n & upper_bound S in IT . (n + 1) ) by A93, A95, A225, A227;
A234: 1 < m + 1 by A225, NAT_1:13;
then A235: IT /. (m + 1) = IT . (m + 1) by A226, FINSEQ_4:15;
A236: n + 1 in dom IT by A226, A234, FINSEQ_3:25;
then A237: IT /. (n + 1) in rng IT by A235, FUNCT_1:def_3;
then A238: IT /. (n + 1) in X by A96;
then A239: IT /. (n + 1) in G \  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A22;
 n + 0 < n + 1 by XREAL_1:6;
then A240: IT /. n <> IT /. (n + 1) by A183, A229, A236;
A241: IT /. (n + 1) is non empty real-bounded interval Subset of REAL by A5, A41, A12, A22, A238;
 IT /. (n + 1) c= union X by A96, A237, ZFMISC_1:74;
then  IT /. (n + 1) c= [.r,s.] by A4, A23;
then A242: IT /. (n + 1) is bounded_above by XXREAL_2:43;
then A243: upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) by A233, A228, A235, SEQ_4:def_1;
hereby  ::_thesis: ( upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) )
assume A244: lower_bound (IT /. n) > lower_bound (IT /. (n + 1)) ; ::_thesis: contradiction
 ( upper_bound (IT /. (n + 1)) = upper_bound (IT /. n) & upper_bound (IT /. n) in IT /. n implies upper_bound (IT /. (n + 1)) in IT /. (n + 1) ) by A22, A79, A210, A183, A202, A204, A206, A227, A229, A230;
then  IT /. n c= IT /. (n + 1) by A232, A241, A243, A244, Th31;
then  IT /. n c< IT /. (n + 1) by A240, XBOOLE_0:def_8;
then  IT /. n in  {  X where X is Subset of (Closed-Interval-TSpace (r,s)) : ( X in G & ex Y being Subset of (Closed-Interval-TSpace (r,s)) st
( Y in G & X c< Y ) )  }  by A10, A231, A239;
hence  contradiction by A22, A230, XBOOLE_0:def_5; ::_thesis: verum
end;
thus  upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) by A233, A228, A235, A242, SEQ_4:def_1; ::_thesis: lower_bound (IT /. (n + 1)) < upper_bound (IT /. n)
percases ( n + 1 = len IT or n + 1 < len IT ) by A226, XXREAL_0:1;
supposeA245: n + 1 = len IT ; ::_thesis: lower_bound (IT /. (n + 1)) < upper_bound (IT /. n)
then  pP < upper_bound (IT /. n) by A202, A206, A233, A228, XXREAL_1:2;
hence  lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) by A202, A204, A206, A245, Th6, XXREAL_1:26; ::_thesis: verum
end;
suppose n + 1 < len IT ; ::_thesis: lower_bound (IT /. (n + 1)) < upper_bound (IT /. n)
then consider a1, b1 being Real such that
 r <= a1 and
A246: a1 < b1 and
 b1 <= s and
A247: IT /. (n + 1) = ].a1,b1.[ by A211, A234;
 a1 < upper_bound (IT /. n) by A233, A228, A235, A247, XXREAL_1:4;
hence  lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) by A246, A247, TOPREAL6:17; ::_thesis: verum
end;
end;
end;
end;
hereby  ::_thesis: ( ( [.r,s.] in F implies IT = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) ) ) )
let n be Nat; ::_thesis: ( 1 <= n implies ( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) & ( n + 2 <= len IT implies not upper_bound (IT /. n) > lower_bound (IT /. (n + 2)) ) ) )
assume A248: 1 <= n ; ::_thesis: ( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) & ( n + 2 <= len IT implies not upper_bound (IT /. n) > lower_bound (IT /. (n + 2)) ) )
thus A249: ( ( n <= len IT implies not IT /. n is empty ) & ( n + 1 <= len IT implies ( lower_bound (IT /. n) <= lower_bound (IT /. (n + 1)) & upper_bound (IT /. n) <= upper_bound (IT /. (n + 1)) & lower_bound (IT /. (n + 1)) < upper_bound (IT /. n) ) ) ) by A224, A248; ::_thesis: ( n + 2 <= len IT implies not upper_bound (IT /. n) > lower_bound (IT /. (n + 2)) )
reconsider m = n as Nat ;
A250: n + 0 < n + 1 by XREAL_1:6;
then A251: 1 < m + 1 by A248, XXREAL_0:2;
assume A252: n + 2 <= len IT ; ::_thesis: not upper_bound (IT /. n) > lower_bound (IT /. (n + 2))
then A253: (n + 1) + 1 <= len IT ;
then A254: m + 1 < len IT by NAT_1:13;
then A255: n + 1 in dom IT by A251, FINSEQ_3:25;
then  IT /. (n + 1) = IT . (n + 1) by PARTFUN1:def_6;
then  IT /. (n + 1) in rng IT by A255, FUNCT_1:def_3;
then A256: IT /. (n + 1) in X by A96;
 0 + 1 <= n + 1 by XREAL_1:6;
then A257: upper_bound (IT /. (n + 1)) <= upper_bound (IT /. ((n + 1) + 1)) by A224, A252;
assume A258: upper_bound (IT /. n) > lower_bound (IT /. (n + 2)) ; ::_thesis: contradiction
consider a1, b1 being Real such that
 r <= a1 and
A259: a1 < b1 and
 b1 <= s and
A260: IT /. (n + 1) = ].a1,b1.[ by A211, A251, A254;
A261: lower_bound ].a1,b1.[ = a1 by A259, TOPREAL6:17;
A262: upper_bound ].a1,b1.[ = b1 by A259, TOPREAL6:17;
A263: IT /. (n + 1) c= (IT /. n) \/ (IT /. (n + 2))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IT /. (n + 1) or x in (IT /. n) \/ (IT /. (n + 2)) )
assume A264: x in IT /. (n + 1) ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
then reconsider x = x as Real ;
A265: a1 < x by A260, A264, XXREAL_1:4;
A266: x < b1 by A260, A264, XXREAL_1:4;
percases ( x < upper_bound (IT /. n) or x >= upper_bound (IT /. n) ) ;
supposeA267: x < upper_bound (IT /. n) ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
percases ( n = 1 or n <> 1 ) ;
supposeA268: n = 1 ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
then  lower_bound (IT /. n) <= x by A4, A49, A94, A100, A256, A264, XXREAL_1:1;
then  x in IT /. n by A38, A50, A49, A94, A100, A267, A268, XXREAL_1:3;
hence  x in (IT /. n) \/ (IT /. (n + 2)) by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA269: n <> 1 ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
 n + 0 < n + 2 by XREAL_1:6;
then A270: n < len IT by A252, XXREAL_0:2;
A271: lower_bound (IT /. n) < x by A249, A253, A260, A261, A265, NAT_1:13, XXREAL_0:2;
 1 < n by A248, A269, XXREAL_0:1;
then consider a, b being Real such that
 r <= a and
A272: a < b and
 b <= s and
A273: IT /. n = ].a,b.[ by A211, A270;
 ( lower_bound (IT /. n) = a & upper_bound (IT /. n) = b ) by A272, A273, TOPREAL6:17;
then  x in IT /. n by A267, A273, A271, XXREAL_1:4;
hence  x in (IT /. n) \/ (IT /. (n + 2)) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
suppose x >= upper_bound (IT /. n) ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
then A274: x > lower_bound (IT /. (n + 2)) by A258, XXREAL_0:2;
percases ( len IT = n + 2 or len IT <> n + 2 ) ;
supposeA275: len IT = n + 2 ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
 x <= s by A4, A256, A264, XXREAL_1:1;
then  x in IT /. (n + 2) by A66, A202, A204, A206, A274, A275, XXREAL_1:2;
hence  x in (IT /. n) \/ (IT /. (n + 2)) by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA276: len IT <> n + 2 ; ::_thesis: x in (IT /. n) \/ (IT /. (n + 2))
 n + 1 < n + 2 by XREAL_1:6;
then A277: 1 < n + 2 by A251, XXREAL_0:2;
 (n + 1) + 1 < len IT by A252, A276, XXREAL_0:1;
then consider a2, b2 being Real such that
 r <= a2 and
A278: a2 < b2 and
 b2 <= s and
A279: IT /. (n + 2) = ].a2,b2.[ by A211, A277;
 upper_bound ].a2,b2.[ = b2 by A278, TOPREAL6:17;
then A280: x < b2 by A257, A260, A262, A266, A279, XXREAL_0:2;
 lower_bound ].a2,b2.[ = a2 by A278, TOPREAL6:17;
then  x in IT /. (n + 2) by A274, A279, A280, XXREAL_1:4;
hence  x in (IT /. n) \/ (IT /. (n + 2)) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
end;
end;
 m + 1 <= m + 2 by XREAL_1:6;
then  1 <= m + 2 by A251, XXREAL_0:2;
then A281: m + 2 in dom IT by A252, FINSEQ_3:25;
then  IT /. (n + 2) = IT . (n + 2) by PARTFUN1:def_6;
then  IT /. (n + 2) in rng IT by A281, FUNCT_1:def_3;
then A282: IT /. (n + 2) in X by A96;
 m <= len IT by A250, A254, XXREAL_0:2;
then A283: n in dom IT by A248, FINSEQ_3:25;
then  IT /. n = IT . n by PARTFUN1:def_6;
then  IT /. n in rng IT by A283, FUNCT_1:def_3;
then A284: IT /. n in X by A96;
 n + 1 < n + 2 by XREAL_1:6;
then A285: IT /. (n + 2) <> IT /. (n + 1) by A183, A255, A281;
 n + 0 < n + 1 by XREAL_1:6;
then  IT /. n <> IT /. (n + 1) by A183, A283, A255;
hence  contradiction by A18, A19, A20, A284, A256, A282, A285, A263, Th48; ::_thesis: verum
end;
thus  ( [.r,s.] in F implies IT = <*[.r,s.]*> ) by A3; ::_thesis: ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) ) )
assume  not [.r,s.] in F ; ::_thesis: ( ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) )
thus  ex p being real number st
( r < p & p <= s & IT . 1 = [.r,p.[ ) ::_thesis: ( ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) ) )
proof
take  kL ; ::_thesis: ( r < kL & kL <= s & IT . 1 = [.r,kL.[ )
thus  r < kL by A38, A48, Th5; ::_thesis: ( kL <= s & IT . 1 = [.r,kL.[ )
 upper_bound LEWY <= upper_bound [.r,s.] by A4, SEQ_4:48;
hence  kL <= s by B4, A47, JORDAN5A:19; ::_thesis: IT . 1 = [.r,kL.[
thus  IT . 1 = [.r,kL.[ by A94; ::_thesis: verum
end;
thus  ex p being real number st
( r <= p & p < s & IT . (len IT) = ].p,s.] ) ::_thesis: for n being Nat st 1 < n & n < len IT holds
ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ )
proof
take  pP ; ::_thesis: ( r <= pP & pP < s & IT . (len IT) = ].pP,s.] )
 lower_bound [.r,s.] <= lower_bound PRAWY by A4, SEQ_4:47;
hence  r <= pP by B4, A66, JORDAN5A:19; ::_thesis: ( pP < s & IT . (len IT) = ].pP,s.] )
thus  pP < s by A64, A67, Th6; ::_thesis: IT . (len IT) = ].pP,s.]
thus  IT . (len IT) = ].pP,s.] by A202, A206; ::_thesis: verum
end;
let n be Nat; ::_thesis: ( 1 < n & n < len IT implies ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ ) )
assume A286: ( 1 < n & n < len IT ) ; ::_thesis: ex p, q being real number st
( r <= p & p < q & q <= s & IT . n = ].p,q.[ )
consider a, b being Real such that
A287: ( r <= a & a < b & b <= s & IT /. n = ].a,b.[ ) by A211, A286;
take  a ; ::_thesis: ex q being real number st
( r <= a & a < q & q <= s & IT . n = ].a,q.[ )
take  b ; ::_thesis: ( r <= a & a < b & b <= s & IT . n = ].a,b.[ )
thus  ( r <= a & a < b & b <= s & IT . n = ].a,b.[ ) by A286, A287, FINSEQ_4:15; ::_thesis: verum
end;
end;
end;
end;

:: deftheorem Def2 defines IntervalCover RCOMP_3:def_2_:_
 for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s)) st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
for b4 being FinSequence of bool REAL holds
( b4 is IntervalCover of F iff ( rng b4 c= F & union (rng b4) = [.r,s.] & ( for n being Nat st 1 <= n holds
( ( n <= len b4 implies not b4 /. n is empty ) & ( n + 1 <= len b4 implies ( lower_bound (b4 /. n) <= lower_bound (b4 /. (n + 1)) & upper_bound (b4 /. n) <= upper_bound (b4 /. (n + 1)) & lower_bound (b4 /. (n + 1)) < upper_bound (b4 /. n) ) ) & ( n + 2 <= len b4 implies upper_bound (b4 /. n) <= lower_bound (b4 /. (n + 2)) ) ) ) & ( [.r,s.] in F implies b4 = <*[.r,s.]*> ) & ( not [.r,s.] in F implies ( ex p being real number st
( r < p & p <= s & b4 . 1 = [.r,p.[ ) & ex p being real number st
( r <= p & p < s & b4 . (len b4) = ].p,s.] ) & ( for n being Nat st 1 < n & n < len b4 holds
ex p, q being real number st
( r <= p & p < q & q <= s & b4 . n = ].p,q.[ ) ) ) ) ) );

theorem :: RCOMP_3:49
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s)) st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & [.r,s.] in F holds
<*[.r,s.]*> is IntervalCover of F
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s)) st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & [.r,s.] in F holds
<*[.r,s.]*> is IntervalCover of F
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & [.r,s.] in F implies <*[.r,s.]*> is IntervalCover of F )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: ( r <= s & [.r,s.] in F ) ; ::_thesis: <*[.r,s.]*> is IntervalCover of F
set f = <*[.r,s.]*>;
A3: for n being Nat st 1 <= n holds
( ( n <= len <*[.r,s.]*> implies not <*[.r,s.]*> /. n is empty ) & ( n + 1 <= len <*[.r,s.]*> implies ( lower_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 1)) & upper_bound (<*[.r,s.]*> /. n) <= upper_bound (<*[.r,s.]*> /. (n + 1)) & lower_bound (<*[.r,s.]*> /. (n + 1)) < upper_bound (<*[.r,s.]*> /. n) ) ) & ( n + 2 <= len <*[.r,s.]*> implies upper_bound (<*[.r,s.]*> /. n) <= lower_bound (<*[.r,s.]*> /. (n + 2)) ) ) by A2, Lm3;
 ( rng <*[.r,s.]*> c= F & union (rng <*[.r,s.]*>) = [.r,s.] ) by A2, Lm3;
hence  <*[.r,s.]*> is IntervalCover of F by A1, A2, A3, Def2; ::_thesis: verum
end;

theorem Th50: :: RCOMP_3:50
for r being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,r))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,r)) & F is open & F is connected holds
C = <*[.r,r.]*>
proof
let r be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,r))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,r)) & F is open & F is connected holds
C = <*[.r,r.]*>
set L = Closed-Interval-TSpace (r,r);
let F be Subset-Family of (Closed-Interval-TSpace (r,r)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,r)) & F is open & F is connected holds
C = <*[.r,r.]*>
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,r)) & F is open & F is connected implies C = <*[.r,r.]*> )
assume that
A1: F is Cover of (Closed-Interval-TSpace (r,r)) and
A2: ( F is open & F is connected ) ; ::_thesis: C = <*[.r,r.]*>
A3: [.r,r.] = {r} by XXREAL_1:17;
 the carrier of (Closed-Interval-TSpace (r,r)) = [.r,r.] by TOPMETR:18;
then  r in the carrier of (Closed-Interval-TSpace (r,r)) by A3, TARSKI:def_1;
then  {r} in F by A1, Th46;
hence  C = <*[.r,r.]*> by A1, A2, A3, Def2; ::_thesis: verum
end;

theorem Th51: :: RCOMP_3:51
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
1 <= len C
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
1 <= len C
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
1 <= len C
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies 1 <= len C )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: 1 <= len C
assume  not 1 <= len C ; ::_thesis: contradiction
then  (len C) + 1 <= 0 + 1 by NAT_1:13;
then A3: C is empty by XREAL_1:6;
 union (rng C) = [.r,s.] by A1, A2, Def2;
hence  contradiction by A2, A3, RELAT_1:38, XXREAL_1:1, ZFMISC_1:2; ::_thesis: verum
end;

theorem Th52: :: RCOMP_3:52
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
C = <*[.r,s.]*>
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
C = <*[.r,s.]*>
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
C = <*[.r,s.]*>
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 implies C = <*[.r,s.]*> )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) and
A2: len C = 1 ; ::_thesis: C = <*[.r,s.]*>
A3: union (rng C) = [.r,s.] by A1, Def2;
 not C is empty by A2;
then  not rng C is empty ;
then  1 in dom C by FINSEQ_3:32;
then A4: C . 1 in rng C by FUNCT_1:def_3;
 C . 1 = [.r,s.]
proof
thus  for a being set st a in C . 1 holds
a in [.r,s.] by A3, A4, TARSKI:def_4; :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: [.r,s.] c= C . 1
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in [.r,s.] or a in C . 1 )
A5: dom C = {1} by A2, FINSEQ_1:2, FINSEQ_1:def_3;
assume  a in [.r,s.] ; ::_thesis: a in C . 1
then consider Z being set  such that
A6: a in Z and
A7: Z in rng C by A3, TARSKI:def_4;
 ex x being set st
( x in dom C & C . x = Z ) by A7, FUNCT_1:def_3;
hence  a in C . 1 by A6, A5, TARSKI:def_1; ::_thesis: verum
end;
hence  C = <*[.r,s.]*> by A2, FINSEQ_1:40; ::_thesis: verum
end;

theorem :: RCOMP_3:53
for r, s being real number
for n, m being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
lower_bound (C /. n) <= lower_bound (C /. m)
proof
let r, s be real number ; ::_thesis: for n, m being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
lower_bound (C /. n) <= lower_bound (C /. m)
let n, m be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
lower_bound (C /. n) <= lower_bound (C /. m)
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
lower_bound (C /. n) <= lower_bound (C /. m)
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m implies lower_bound (C /. n) <= lower_bound (C /. m) )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) and
A2: n in dom C and
A3: ( m in dom C & n < m ) ; ::_thesis: lower_bound (C /. n) <= lower_bound (C /. m)
defpred S2[ Nat] means ( $1 in dom C & n < $1 implies lower_bound (C /. n) <= lower_bound (C /. $1) );
A4: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume that
A5: S2[k] and
A6: k + 1 in dom C and
A7: n < k + 1 ; ::_thesis: lower_bound (C /. n) <= lower_bound (C /. (k + 1))
percases ( k = 0 or k in dom C ) by A6, TOPREALA:2;
suppose k = 0 ; ::_thesis: lower_bound (C /. n) <= lower_bound (C /. (k + 1))
then  n = 0 by A7, NAT_1:13;
hence  lower_bound (C /. n) <= lower_bound (C /. (k + 1)) by A2, FINSEQ_3:24; ::_thesis: verum
end;
supposeA8: k in dom C ; ::_thesis: lower_bound (C /. n) <= lower_bound (C /. (k + 1))
A9: k + 1 <= len C by A6, FINSEQ_3:25;
A10: n <= k by A7, NAT_1:13;
 1 <= k by A8, FINSEQ_3:25;
then  lower_bound (C /. k) <= lower_bound (C /. (k + 1)) by A1, A9, Def2;
hence  lower_bound (C /. n) <= lower_bound (C /. (k + 1)) by A5, A8, A10, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A11: S2[ 0 ] ;
 for k being Nat holds S2[k] from NAT_1:sch_2(A11, A4);
hence  lower_bound (C /. n) <= lower_bound (C /. m) by A3; ::_thesis: verum
end;

theorem :: RCOMP_3:54
for r, s being real number
for n, m being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
upper_bound (C /. n) <= upper_bound (C /. m)
proof
let r, s be real number ; ::_thesis: for n, m being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
upper_bound (C /. n) <= upper_bound (C /. m)
let n, m be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
upper_bound (C /. n) <= upper_bound (C /. m)
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m holds
upper_bound (C /. n) <= upper_bound (C /. m)
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & n in dom C & m in dom C & n < m implies upper_bound (C /. n) <= upper_bound (C /. m) )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) and
A2: n in dom C and
A3: ( m in dom C & n < m ) ; ::_thesis: upper_bound (C /. n) <= upper_bound (C /. m)
defpred S2[ Nat] means ( $1 in dom C & n < $1 implies upper_bound (C /. n) <= upper_bound (C /. $1) );
A4: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume that
A5: S2[k] and
A6: k + 1 in dom C and
A7: n < k + 1 ; ::_thesis: upper_bound (C /. n) <= upper_bound (C /. (k + 1))
percases ( k = 0 or k in dom C ) by A6, TOPREALA:2;
suppose k = 0 ; ::_thesis: upper_bound (C /. n) <= upper_bound (C /. (k + 1))
then  n = 0 by A7, NAT_1:13;
hence  upper_bound (C /. n) <= upper_bound (C /. (k + 1)) by A2, FINSEQ_3:24; ::_thesis: verum
end;
supposeA8: k in dom C ; ::_thesis: upper_bound (C /. n) <= upper_bound (C /. (k + 1))
A9: k + 1 <= len C by A6, FINSEQ_3:25;
A10: n <= k by A7, NAT_1:13;
 1 <= k by A8, FINSEQ_3:25;
then  upper_bound (C /. k) <= upper_bound (C /. (k + 1)) by A1, A9, Def2;
hence  upper_bound (C /. n) <= upper_bound (C /. (k + 1)) by A5, A8, A10, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A11: S2[ 0 ] ;
 for k being Nat holds S2[k] from NAT_1:sch_2(A11, A4);
hence  upper_bound (C /. n) <= upper_bound (C /. m) by A3; ::_thesis: verum
end;

theorem Th55: :: RCOMP_3:55
for r, s being real number
for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
proof
let r, s be real number ; ::_thesis: for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let n be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C holds
not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C implies not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty )
assume  ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 <= len C ) ; ::_thesis: not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty
then  lower_bound (C /. (n + 1)) < upper_bound (C /. n) by Def2;
hence  not ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ is empty by XXREAL_1:33; ::_thesis: verum
end;

theorem :: RCOMP_3:56
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
lower_bound (C /. 1) = r
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
lower_bound (C /. 1) = r
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
lower_bound (C /. 1) = r
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies lower_bound (C /. 1) = r )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: lower_bound (C /. 1) = r
 1 <= len C by A1, A2, Th51;
then A3: C . 1 = C /. 1 by FINSEQ_4:15;
percases ( [.r,s.] in F or not [.r,s.] in F ) ;
suppose [.r,s.] in F ; ::_thesis: lower_bound (C /. 1) = r
then  C = <*[.r,s.]*> by A1, A2, Def2;
then  C /. 1 = [.r,s.] by FINSEQ_4:16;
hence  lower_bound (C /. 1) = r by A2, JORDAN5A:19; ::_thesis: verum
end;
suppose not [.r,s.] in F ; ::_thesis: lower_bound (C /. 1) = r
then  ex p being real number st
( r < p & p <= s & C . 1 = [.r,p.[ ) by A1, A2, Def2;
hence  lower_bound (C /. 1) = r by A3, Th4; ::_thesis: verum
end;
end;
end;

theorem Th57: :: RCOMP_3:57
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
r in C /. 1
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
r in C /. 1
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
r in C /. 1
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies r in C /. 1 )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: r in C /. 1
 1 <= len C by A1, A2, Th51;
then A3: C . 1 = C /. 1 by FINSEQ_4:15;
percases ( [.r,s.] in F or not [.r,s.] in F ) ;
suppose [.r,s.] in F ; ::_thesis: r in C /. 1
then  C = <*[.r,s.]*> by A1, A2, Def2;
then  C /. 1 = [.r,s.] by FINSEQ_4:16;
hence  r in C /. 1 by A2, XXREAL_1:1; ::_thesis: verum
end;
suppose not [.r,s.] in F ; ::_thesis: r in C /. 1
then  ex p being real number st
( r < p & p <= s & C . 1 = [.r,p.[ ) by A1, A2, Def2;
hence  r in C /. 1 by A3, XXREAL_1:3; ::_thesis: verum
end;
end;
end;

theorem :: RCOMP_3:58
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
upper_bound (C /. (len C)) = s
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
upper_bound (C /. (len C)) = s
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
upper_bound (C /. (len C)) = s
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies upper_bound (C /. (len C)) = s )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: upper_bound (C /. (len C)) = s
 1 <= len C by A1, A2, Th51;
then A3: C . (len C) = C /. (len C) by FINSEQ_4:15;
percases ( [.r,s.] in F or not [.r,s.] in F ) ;
suppose [.r,s.] in F ; ::_thesis: upper_bound (C /. (len C)) = s
then  C = <*[.r,s.]*> by A1, A2, Def2;
then  ( C /. 1 = [.r,s.] & len C = 1 ) by FINSEQ_1:39, FINSEQ_4:16;
hence  upper_bound (C /. (len C)) = s by A2, JORDAN5A:19; ::_thesis: verum
end;
suppose not [.r,s.] in F ; ::_thesis: upper_bound (C /. (len C)) = s
then  ex p being real number st
( r <= p & p < s & C . (len C) = ].p,s.] ) by A1, A2, Def2;
hence  upper_bound (C /. (len C)) = s by A3, Th7; ::_thesis: verum
end;
end;
end;

theorem Th59: :: RCOMP_3:59
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
s in C /. (len C)
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
s in C /. (len C)
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
s in C /. (len C)
let C be IntervalCover of F; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies s in C /. (len C) )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: s in C /. (len C)
 1 <= len C by A1, A2, Th51;
then A3: C . (len C) = C /. (len C) by FINSEQ_4:15;
percases ( [.r,s.] in F or not [.r,s.] in F ) ;
suppose [.r,s.] in F ; ::_thesis: s in C /. (len C)
then  C = <*[.r,s.]*> by A1, A2, Def2;
then  ( C /. 1 = [.r,s.] & len C = 1 ) by FINSEQ_1:39, FINSEQ_4:16;
hence  s in C /. (len C) by A2, XXREAL_1:1; ::_thesis: verum
end;
suppose not [.r,s.] in F ; ::_thesis: s in C /. (len C)
then  ex p being real number st
( r <= p & p < s & C . (len C) = ].p,s.] ) by A1, A2, Def2;
hence  s in C /. (len C) by A3, XXREAL_1:2; ::_thesis: verum
end;
end;
end;

definition
let r, s be real number ;
let F be Subset-Family of (Closed-Interval-TSpace (r,s));
let C be IntervalCover of F;
assume B1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) ;
mode IntervalCoverPts of C ->  FinSequence of REAL  means :Def3: :: RCOMP_3:def 3
( len it = (len C) + 1 & it . 1 = r & it . (len it) = s & ( for n being Nat st 1 <= n & n + 1 < len it holds
it . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) );
existence
 ex b1 being FinSequence of REAL st
( len b1 = (len C) + 1 & b1 . 1 = r & b1 . (len b1) = s & ( for n being Nat st 1 <= n & n + 1 < len b1 holds
b1 . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) )
proof
set A = (len C) + 1;
defpred S2[ Nat, set ] means ( ( $1 = 1 implies $2 = r ) & ( $1 = (len C) + 1 implies $2 = s ) & ( 2 <= $1 & $1 <= len C implies $2 in ].(lower_bound (C /. $1)),(upper_bound (C /. ($1 - 1))).[ ) );
A1: 0 + 1 <= len C by B1, Th51;
then A2: 0 + 1 < (len C) + 1 by XREAL_1:6;
A3: for k being Nat st k in Seg ((len C) + 1) holds
ex x being Element of REAL st S2[k,x]
proof
reconsider r = r, s = s as Real by XREAL_0:def_1;
let k be Nat; ::_thesis: ( k in Seg ((len C) + 1) implies ex x being Element of REAL st S2[k,x] )
A4: (len C) + 0 < (len C) + 1 by XREAL_1:6;
assume  k in Seg ((len C) + 1) ; ::_thesis: ex x being Element of REAL st S2[k,x]
then A5: ( 1 <= k & k <= (len C) + 1 ) by FINSEQ_1:1;
percases ( k = 1 or k = (len C) + 1 or ( 1 < k & k < (len C) + 1 ) ) by A5, XXREAL_0:1;
supposeA6: k = 1 ; ::_thesis: ex x being Element of REAL st S2[k,x]
take  r ; ::_thesis: S2[k,r]
thus  S2[k,r] by A1, A6; ::_thesis: verum
end;
supposeA7: k = (len C) + 1 ; ::_thesis: ex x being Element of REAL st S2[k,x]
take  s ; ::_thesis: S2[k,s]
thus  S2[k,s] by A1, A4, A7; ::_thesis: verum
end;
supposethat A8: 1 < k and
A9: k < (len C) + 1 ; ::_thesis: ex x being Element of REAL st S2[k,x]
A10: k - 1 in NAT by A8, INT_1:5;
A11: k <= len C by A9, NAT_1:13;
 1 - 1 < k - 1 by A8, XREAL_1:14;
then  0 + 1 <= k - 1 by A10, NAT_1:13;
then  not ].(lower_bound (C /. ((k - 1) + 1))),(upper_bound (C /. (k - 1))).[ is empty by B1, A10, A11, Th55;
then consider x being set  such that
A12: x in ].(lower_bound (C /. ((k - 1) + 1))),(upper_bound (C /. (k - 1))).[ by XBOOLE_0:def_1;
reconsider x = x as Real by A12;
take  x ; ::_thesis: S2[k,x]
thus  S2[k,x] by A8, A9, A12; ::_thesis: verum
end;
end;
end;
consider p being FinSequence of REAL  such that
A13: dom p = Seg ((len C) + 1) and
A14: for k being Nat st k in Seg ((len C) + 1) holds
S2[k,p . k] from FINSEQ_1:sch_5(A3);
take  p ; ::_thesis: ( len p = (len C) + 1 & p . 1 = r & p . (len p) = s & ( for n being Nat st 1 <= n & n + 1 < len p holds
p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) )
thus A15: len p = (len C) + 1 by A13, FINSEQ_1:def_3; ::_thesis: ( p . 1 = r & p . (len p) = s & ( for n being Nat st 1 <= n & n + 1 < len p holds
p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) )
 1 in Seg ((len C) + 1) by A2, FINSEQ_1:1;
hence  p . 1 = r by A14; ::_thesis: ( p . (len p) = s & ( for n being Nat st 1 <= n & n + 1 < len p holds
p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) )
 len p in Seg ((len C) + 1) by A2, A15, FINSEQ_1:1;
hence  p . (len p) = s by A14, A15; ::_thesis: for n being Nat st 1 <= n & n + 1 < len p holds
p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[
let n be Nat; ::_thesis: ( 1 <= n & n + 1 < len p implies p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ )
assume  1 <= n ; ::_thesis: ( not n + 1 < len p or p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ )
then A16: 1 + 1 <= n + 1 by XREAL_1:6;
assume A17: n + 1 < len p ; ::_thesis: p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[
 0 + 1 <= n + 1 by XREAL_1:6;
then A18: n + 1 in Seg ((len C) + 1) by A15, A17, FINSEQ_1:1;
 n + 1 <= len C by A15, A17, NAT_1:13;
then  p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. ((n + 1) - 1))).[ by A14, A18, A16;
hence  p . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines IntervalCoverPts RCOMP_3:def_3_:_
 for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
for b5 being FinSequence of REAL holds
( b5 is IntervalCoverPts of C iff ( len b5 = (len C) + 1 & b5 . 1 = r & b5 . (len b5) = s & ( for n being Nat st 1 <= n & n + 1 < len b5 holds
b5 . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ ) ) );

theorem Th60: :: RCOMP_3:60
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
2 <= len G
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
2 <= len G
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
2 <= len G
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s holds
2 <= len G
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s implies 2 <= len G )
assume A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) ; ::_thesis: 2 <= len G
then  1 <= len C by Th51;
then  1 + 1 <= (len C) + 1 by XREAL_1:6;
hence  2 <= len G by A1, Def3; ::_thesis: verum
end;

theorem Th61: :: RCOMP_3:61
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
G = <*r,s*>
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
G = <*r,s*>
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
G = <*r,s*>
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 holds
G = <*r,s*>
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & len C = 1 implies G = <*r,s*> )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) and
A2: len C = 1 ; ::_thesis: G = <*r,s*>
A3: G . 1 = r by A1, Def3;
A4: len G = (len C) + 1 by A1, Def3;
then  G . 2 = s by A1, A2, Def3;
hence  G = <*r,s*> by A2, A4, A3, FINSEQ_1:44; ::_thesis: verum
end;

theorem Th62: :: RCOMP_3:62
for r, s being real number
for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
proof
let r, s be real number ; ::_thesis: for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let n be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G holds
G . (n + 1) < upper_bound (C /. n)
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G implies G . (n + 1) < upper_bound (C /. n) )
assume  ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n + 1 < len G ) ; ::_thesis: G . (n + 1) < upper_bound (C /. n)
then  G . (n + 1) in ].(lower_bound (C /. (n + 1))),(upper_bound (C /. n)).[ by Def3;
hence  G . (n + 1) < upper_bound (C /. n) by XXREAL_1:4; ::_thesis: verum
end;

theorem Th63: :: RCOMP_3:63
for r, s being real number
for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C holds
lower_bound (C /. n) < G . n
proof
let r, s be real number ; ::_thesis: for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C holds
lower_bound (C /. n) < G . n
let n be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C holds
lower_bound (C /. n) < G . n
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C holds
lower_bound (C /. n) < G . n
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C holds
lower_bound (C /. n) < G . n
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 < n & n <= len C implies lower_bound (C /. n) < G . n )
set w = n -' 1;
assume A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s ) ; ::_thesis: ( not 1 < n or not n <= len C or lower_bound (C /. n) < G . n )
then A2: len G = (len C) + 1 by Def3;
assume that
A3: 1 < n and
A4: n <= len C ; ::_thesis: lower_bound (C /. n) < G . n
A5: n < (len C) + 1 by A4, NAT_1:13;
 1 - 1 <= n - 1 by A3, XREAL_1:9;
then A6: n -' 1 = n - 1 by XREAL_0:def_2;
then  n = (n -' 1) + 1 ;
then  1 <= n -' 1 by A3, NAT_1:13;
then  G . ((n -' 1) + 1) in ].(lower_bound (C /. ((n -' 1) + 1))),(upper_bound (C /. (n -' 1))).[ by A1, A2, A6, A5, Def3;
hence  lower_bound (C /. n) < G . n by A6, XXREAL_1:4; ::_thesis: verum
end;

theorem Th64: :: RCOMP_3:64
for r, s being real number
for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C holds
G . n <= lower_bound (C /. (n + 1))
proof
let r, s be real number ; ::_thesis: for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C holds
G . n <= lower_bound (C /. (n + 1))
let n be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C holds
G . n <= lower_bound (C /. (n + 1))
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C holds
G . n <= lower_bound (C /. (n + 1))
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C holds
G . n <= lower_bound (C /. (n + 1))
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len C implies G . n <= lower_bound (C /. (n + 1)) )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r <= s ; ::_thesis: ( not 1 <= n or not n < len C or G . n <= lower_bound (C /. (n + 1)) )
set w = n -' 1;
assume that
A3: 1 <= n and
A4: n < len C ; ::_thesis: G . n <= lower_bound (C /. (n + 1))
A5: n + 1 <= len C by A4, NAT_1:13;
percases ( n = 1 or 1 < n ) by A3, XXREAL_0:1;
supposeA6: n = 1 ; ::_thesis: G . n <= lower_bound (C /. (n + 1))
 0 + 1 <= n + 1 by XREAL_1:6;
then A7: not C /. (n + 1) is empty by A1, A2, A5, Def2;
A8: G . 1 = r by A1, A2, Def3;
A9: rng C c= F by A1, A2, Def2;
 1 + 1 <= len C by A4, A6, NAT_1:13;
then A10: 2 in dom C by FINSEQ_3:25;
then  C . 2 in rng C by FUNCT_1:def_3;
then  C . 2 in F by A9;
then  C /. 2 in F by A10, PARTFUN1:def_6;
then  C /. (n + 1) c= the carrier of (Closed-Interval-TSpace (r,s)) by A6;
then A11: C /. (n + 1) c= [.r,s.] by A2, TOPMETR:18;
then  C /. (n + 1) is bounded_below by XXREAL_2:44;
then  lower_bound (C /. (n + 1)) in [.r,s.] by A7, A11, Th1;
hence  G . n <= lower_bound (C /. (n + 1)) by A6, A8, XXREAL_1:1; ::_thesis: verum
end;
suppose 1 < n ; ::_thesis: G . n <= lower_bound (C /. (n + 1))
then A12: 1 - 1 < n - 1 by XREAL_1:9;
then A13: n -' 1 = n - 1 by XREAL_0:def_2;
then A14: 0 + 1 <= n -' 1 by A12, NAT_1:13;
 len G = (len C) + 1 by A1, A2, Def3;
then A15: n + 1 < ((len G) - 1) + 1 by A4, XREAL_1:6;
 n - 1 < n - 0 by XREAL_1:15;
then  (n -' 1) + 1 < n + 1 by A13, XREAL_1:6;
then  (n -' 1) + 1 < len G by A15, XXREAL_0:2;
then A16: G . ((n -' 1) + 1) < upper_bound (C /. (n -' 1)) by A1, A2, A14, Th62;
 n + 1 <= len C by A4, NAT_1:13;
then  upper_bound (C /. (n -' 1)) <= lower_bound (C /. ((n -' 1) + 2)) by A1, A2, A13, A14, Def2;
hence  G . n <= lower_bound (C /. (n + 1)) by A13, A16, XXREAL_0:2; ::_thesis: verum
end;
end;
end;

theorem Th65: :: RCOMP_3:65
for r, s being real number
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r < s holds
G is increasing
proof
let r, s be real number ; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r < s holds
G is increasing
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r < s holds
G is increasing
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r < s holds
G is increasing
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r < s implies G is increasing )
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected ) and
A2: r < s ; ::_thesis: G is increasing
let m, n be Element of NAT ; :: according to SEQM_3:def_1 ::_thesis: ( not m in dom G or not n in dom G or n <= m or not K616(G,n) <= K616(G,m) )
assume A3: ( m in dom G & n in dom G & m < n ) ; ::_thesis: not K616(G,n) <= K616(G,m)
defpred S2[ Nat] means ( m < $1 & m in dom G & $1 in dom G implies G . m < G . $1 );
A4: for k being Nat st S2[k] holds
S2[k + 1]
proof
A5: len G = (len C) + 1 by A1, A2, Def3;
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume that
A6: S2[k] and
A7: m < k + 1 and
A8: m in dom G and
A9: k + 1 in dom G ; ::_thesis: G . m < G . (k + 1)
A10: 1 <= m by A8, FINSEQ_3:25;
A11: k + 1 <= len G by A9, FINSEQ_3:25;
 k + 0 <= k + 1 by XREAL_1:6;
then A12: k <= len G by A11, XXREAL_0:2;
A13: m <= k by A7, NAT_1:13;
then A14: 1 <= k by A10, XXREAL_0:2;
percases ( ( 1 < k & k + 1 < len G ) or k = 1 or k + 1 = len G ) by A14, A11, XXREAL_0:1;
supposethat A15: 1 < k and
A16: k + 1 < len G ; ::_thesis: G . m < G . (k + 1)
 G . (k + 1) in ].(lower_bound (C /. (k + 1))),(upper_bound (C /. k)).[ by A1, A2, A15, A16, Def3;
then A17: lower_bound (C /. (k + 1)) < G . (k + 1) by XXREAL_1:4;
 k < len C by A5, A16, XREAL_1:6;
then  G . k <= lower_bound (C /. (k + 1)) by A1, A2, A15, Th64;
then  G . k < G . (k + 1) by A17, XXREAL_0:2;
hence  G . m < G . (k + 1) by A6, A8, A13, A12, A15, FINSEQ_3:25, XXREAL_0:1, XXREAL_0:2; ::_thesis: verum
end;
supposeA18: k = 1 ; ::_thesis: G . m < G . (k + 1)
A19: 1 <= len C by A1, A2, Th51;
A20: m <= 1 by A7, A18, NAT_1:13;
percases ( 1 < len C or 1 = len C ) by A19, XXREAL_0:1;
supposeA21: 1 < len C ; ::_thesis: G . m < G . (k + 1)
then  1 + 1 <= len C by NAT_1:13;
then A22: lower_bound (C /. 2) < G . 2 by A1, A2, Th63;
 G . 1 <= lower_bound (C /. (1 + 1)) by A1, A2, A21, Th64;
then  G . 1 < G . 2 by A22, XXREAL_0:2;
hence  G . m < G . (k + 1) by A10, A18, A20, XXREAL_0:1; ::_thesis: verum
end;
suppose 1 = len C ; ::_thesis: G . m < G . (k + 1)
then  G = <*r,s*> by A1, A2, Th61;
then  ( G . 1 = r & G . 2 = s ) by FINSEQ_1:44;
hence  G . m < G . (k + 1) by A2, A10, A18, A20, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
supposeA23: k + 1 = len G ; ::_thesis: G . m < G . (k + 1)
then A24: G . (k + 1) = s by A1, A2, Def3;
percases ( 1 < m or m = 1 ) by A10, XXREAL_0:1;
supposeA25: 1 < m ; ::_thesis: G . m < G . (k + 1)
set z = m -' 1;
 1 - 1 <= m - 1 by A10, XREAL_1:9;
then A26: m -' 1 = m - 1 by XREAL_0:def_2;
then A27: (m -' 1) + 1 < len G by A7, A23;
then A28: m -' 1 <= len C by A5, XREAL_1:6;
 1 + 1 <= m by A25, NAT_1:13;
then A29: (1 + 1) - 1 <= m - 1 by XREAL_1:9;
then A30: 1 <= m -' 1 by XREAL_0:def_2;
then A31: not C /. (m -' 1) is empty by A1, A2, A28, Def2;
A32: rng C c= F by A1, A2, Def2;
A33: m -' 1 in dom C by A30, A28, FINSEQ_3:25;
then  C . (m -' 1) in rng C by FUNCT_1:def_3;
then  C . (m -' 1) in F by A32;
then  C /. (m -' 1) in F by A33, PARTFUN1:def_6;
then  C /. (m -' 1) c= the carrier of (Closed-Interval-TSpace (r,s)) ;
then A34: C /. (m -' 1) c= [.r,s.] by A2, TOPMETR:18;
then  C /. (m -' 1) is bounded_above by XXREAL_2:43;
then  upper_bound (C /. (m -' 1)) in [.r,s.] by A34, A31, Th2;
then A35: upper_bound (C /. (m -' 1)) <= s by XXREAL_1:1;
 G . m < upper_bound (C /. (m -' 1)) by A1, A2, A26, A29, A27, Th62;
hence  G . m < G . (k + 1) by A24, A35, XXREAL_0:2; ::_thesis: verum
end;
suppose m = 1 ; ::_thesis: G . m < G . (k + 1)
hence  G . m < G . (k + 1) by A1, A2, A24, Def3; ::_thesis: verum
end;
end;
end;
end;
end;
A36: S2[ 0 ] ;
 for k being Nat holds S2[k] from NAT_1:sch_2(A36, A4);
hence  not K616(G,n) <= K616(G,m) by A3; ::_thesis: verum
end;

theorem :: RCOMP_3:66
for r, s being real number
for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G holds
[.(G . n),(G . (n + 1)).] c= C . n
proof
let r, s be real number ; ::_thesis: for n being Nat
for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G holds
[.(G . n),(G . (n + 1)).] c= C . n
let n be Nat; ::_thesis: for F being Subset-Family of (Closed-Interval-TSpace (r,s))
for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G holds
[.(G . n),(G . (n + 1)).] c= C . n
let F be Subset-Family of (Closed-Interval-TSpace (r,s)); ::_thesis: for C being IntervalCover of F
for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G holds
[.(G . n),(G . (n + 1)).] c= C . n
let C be IntervalCover of F; ::_thesis: for G being IntervalCoverPts of C st F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G holds
[.(G . n),(G . (n + 1)).] c= C . n
let G be IntervalCoverPts of C; ::_thesis: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open & F is connected & r <= s & 1 <= n & n < len G implies [.(G . n),(G . (n + 1)).] c= C . n )
set L = Closed-Interval-TSpace (r,s);
assume that
A1: ( F is Cover of (Closed-Interval-TSpace (r,s)) & F is open ) and
A2: F is connected and
A3: r <= s and
A4: 1 <= n and
A5: n < len G ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
A6: len G = (len C) + 1 by A1, A2, A3, Def3;
then A7: n <= len C by A5, NAT_1:13;
then A8: C /. n = C . n by A4, FINSEQ_4:15;
 n in dom C by A4, A7, FINSEQ_3:25;
then A9: C . n in rng C by FUNCT_1:def_3;
 rng C c= F by A1, A2, A3, Def2;
then  C /. n in F by A8, A9;
then  C /. n is connected Subset of (Closed-Interval-TSpace (r,s)) by A2, Def1;
then A10: C /. n is interval by Th43;
A11: not C /. n is empty by A1, A2, A3, A4, A7, Def2;
A12: n + 1 <= len G by A5, NAT_1:13;
 0 + 1 <= n + 1 by XREAL_1:6;
then A13: n + 1 in dom G by A12, FINSEQ_3:25;
A14: n in dom G by A4, A5, FINSEQ_3:25;
A15: n + 0 < n + 1 by XREAL_1:6;
percases ( r = s or r < s ) by A3, XXREAL_0:1;
suppose r = s ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
then  C = <*[.r,r.]*> by A1, A2, Th50;
then A16: len C = 1 by FINSEQ_1:40;
then  G = <*r,s*> by A1, A2, A3, Th61;
then A17: ( G . 1 = r & G . 2 = s ) by FINSEQ_1:44;
 ( n = 1 & C = <*[.r,s.]*> ) by A1, A2, A3, A4, A7, A16, Th52, XXREAL_0:1;
hence  [.(G . n),(G . (n + 1)).] c= C . n by A17, FINSEQ_1:40; ::_thesis: verum
end;
suppose r < s ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
then  G is increasing by A1, A2, Th65;
then A18: G . n < G . (n + 1) by A14, A13, A15, SEQM_3:def_1;
A19: 2 <= len G by A1, A2, A3, Th60;
percases ( ( n = 1 & len G = 2 ) or ( n = 1 & 1 + 1 < len G ) or ( 1 < n & len G = n + 1 ) or ( 1 < n & n + 1 < len G ) ) by A4, A12, A19, XXREAL_0:1;
supposethat A20: n = 1 and
A21: len G = 2 ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
 G = <*r,s*> by A1, A2, A3, A6, A21, Th61;
then A22: ( G . 1 = r & G . 2 = s ) by FINSEQ_1:44;
 C = <*[.r,s.]*> by A1, A2, A3, A6, A21, Th52;
hence  [.(G . n),(G . (n + 1)).] c= C . n by A20, A22, FINSEQ_1:40; ::_thesis: verum
end;
supposethat A23: n = 1 and
A24: 1 + 1 < len G ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
 G . (1 + 1) in ].(lower_bound (C /. (1 + 1))),(upper_bound (C /. 1)).[ by A1, A2, A3, A24, Def3;
then A25: lower_bound (C /. (1 + 1)) < G . 2 by XXREAL_1:4;
 1 + 1 <= len C by A6, A24, NAT_1:13;
then  lower_bound (C /. 1) <= lower_bound (C /. (1 + 1)) by A1, A2, A3, Def2;
then A26: lower_bound (C /. n) < G . (n + 1) by A23, A25, XXREAL_0:2;
A27: ( G . 1 = r & r in C /. 1 ) by A1, A2, A3, Def3, Th57;
 G . (n + 1) < upper_bound (C /. n) by A1, A2, A3, A23, A24, Th62;
then  G . (n + 1) in C . n by A8, A10, A11, A26, Th30;
hence  [.(G . n),(G . (n + 1)).] c= C . n by A8, A10, A23, A27, XXREAL_2:def_12; ::_thesis: verum
end;
supposethat A28: 1 < n and
A29: len G = n + 1 ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
 1 - 1 < n - 1 by A28, XREAL_1:9;
then A30: ( 0 + 1 <= n - 1 & n - 1 is Element of NAT ) by INT_1:3, INT_1:7;
then  G . ((n - 1) + 1) in ].(lower_bound (C /. ((n - 1) + 1))),(upper_bound (C /. (n - 1))).[ by A1, A2, A3, A15, A29, Def3;
then A31: G . n < upper_bound (C /. (n - 1)) by XXREAL_1:4;
 upper_bound (C /. (n - 1)) <= upper_bound (C /. ((n - 1) + 1)) by A1, A2, A3, A6, A29, A30, Def2;
then A32: G . n < upper_bound (C /. n) by A31, XXREAL_0:2;
 G . (len G) = s by A1, A2, A3, Def3;
then A33: G . (n + 1) in C . n by A1, A2, A3, A6, A8, A29, Th59;
 lower_bound (C /. n) < G . n by A1, A2, A3, A6, A28, A29, Th63;
then  G . n in C . n by A8, A10, A11, A32, Th30;
hence  [.(G . n),(G . (n + 1)).] c= C . n by A8, A10, A33, XXREAL_2:def_12; ::_thesis: verum
end;
supposethat A34: 1 < n and
A35: n + 1 < len G ; ::_thesis: [.(G . n),(G . (n + 1)).] c= C . n
A36: G . (n + 1) < upper_bound (C /. n) by A1, A2, A3, A4, A35, Th62;
 n <= len C by A5, A6, NAT_1:13;
then A37: lower_bound (C /. n) < G . n by A1, A2, A3, A34, Th63;
then  lower_bound (C /. n) < G . (n + 1) by A18, XXREAL_0:2;
then A38: G . (n + 1) in C . n by A8, A10, A11, A36, Th30;
 G . n < upper_bound (C /. n) by A18, A36, XXREAL_0:2;
then  G . n in C . n by A8, A10, A11, A37, Th30;
hence  [.(G . n),(G . (n + 1)).] c= C . n by A8, A10, A38, XXREAL_2:def_12; ::_thesis: verum
end;
end;
end;
end;
end;