:: CHAIN_1 semantic presentation

begin


theorem Th1: :: CHAIN_1:1
for x, y being real number st x < y holds
ex z being Real st
( x < z & z < y )
proof
let x, y be real number ; ::_thesis: ( x < y implies ex z being Real st
( x < z & z < y ) )
assume  x < y ; ::_thesis: ex z being Real st
( x < z & z < y )
then consider z being real number  such that
A1: x < z and
A2: z < y by XREAL_1:5;
reconsider z = z as Real by XREAL_0:def_1;
take  z ; ::_thesis: ( x < z & z < y )
thus  ( x < z & z < y ) by A1, A2; ::_thesis: verum
end;

theorem Th2: :: CHAIN_1:2
for x, y being real number ex z being Real st
( x < z & y < z )
proof
let x, y be real number ; ::_thesis: ex z being Real st
( x < z & y < z )
reconsider x = x, y = y as Real by XREAL_0:def_1;
take z = (max (x,y)) + 1; ::_thesis: ( x < z & y < z )
A1: x + 0 < z by XREAL_1:8, XXREAL_0:25;
 y + 0 < z by XREAL_1:8, XXREAL_0:25;
hence  ( x < z & y < z ) by A1; ::_thesis: verum
end;

scheme :: CHAIN_1:sch 1
FrSet12{ F1() ->  non empty set , F2() ->  non empty set , P1[ set , set ], F3( set , set ) ->  Element of F1() } :
  {  F3(x,y) where x, y is Element of F2() : P1[x,y]  }  c= F1()
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in  {  F3(x,y) where x, y is Element of F2() : P1[x,y]  }  or z in F1() )
assume  z in  {  F3(x,y) where x, y is Element of F2() : P1[x,y]  }  ; ::_thesis: z in F1()
then  ex x, y being Element of F2() st
( z = F3(x,y) & P1[x,y] ) ;
hence  z in F1() ; ::_thesis: verum
end;

definition
let B be set ;
let A be Subset of B;
:: original: bool
redefine func bool A ->  Subset-Family of B;
coherence
 bool A is Subset-Family of B by ZFMISC_1:67;
end;

definition
let d be real Element of NAT ;
redefine attr d is empty  means :: CHAIN_1:def 1
 not d > 0 ;
compatibility
 ( d is zero iff not d > 0 ) ;
end;

:: deftheorem  defines zero CHAIN_1:def_1_:_
 for d being real Element of NAT holds
( d is zero iff not d > 0 );

definition
let d be Element of NAT ;
redefine attr d is empty  means :Def2: :: CHAIN_1:def 2
 not d >= 1;
compatibility
 ( d is zero iff not d >= 1 )
proof
 ( not d is zero iff d >= 1 + 0 ) by NAT_1:13;
hence  ( d is zero iff not d >= 1 ) ; ::_thesis: verum
end;
end;

:: deftheorem Def2 defines zero CHAIN_1:def_2_:_
 for d being Element of NAT holds
( d is zero iff not d >= 1 );


theorem Th3: :: CHAIN_1:3
for x, y being set holds
( {x,y} is trivial iff x = y )
proof
let x, y be set ; ::_thesis: ( {x,y} is trivial iff x = y )
hereby  ::_thesis: ( x = y implies {x,y} is trivial )
A1: x in {x,y} by TARSKI:def_2;
 y in {x,y} by TARSKI:def_2;
hence  ( {x,y} is trivial implies x = y ) by A1, ZFMISC_1:def_10; ::_thesis: verum
end;
 {x,x} = {x} by ENUMSET1:29;
hence  ( x = y implies {x,y} is trivial ) ; ::_thesis: verum
end;

registration
cluster non trivial finite for set ;
existence
 ex b1 being set st
( not b1 is trivial & b1 is finite )
proof
take  {0,1} ; ::_thesis: ( not {0,1} is trivial & {0,1} is finite )
thus  ( not {0,1} is trivial & {0,1} is finite ) by Th3; ::_thesis: verum
end;
end;

registration
let X be non trivial set ;
let Y be set ;
clusterX \/ Y ->  non trivial ;
coherence
 not X \/ Y is trivial
proof
consider x, y being set  such that
A1: x in X and
A2: y in X and
A3: x <> y by ZFMISC_1:def_10;
take  x ; :: according to ZFMISC_1:def_10 ::_thesis: ex b1 being set st
( x in X \/ Y & b1 in X \/ Y & not x = b1 )
take  y ; ::_thesis: ( x in X \/ Y & y in X \/ Y & not x = y )
thus  ( x in X \/ Y & y in X \/ Y & not x = y ) by A1, A2, A3, XBOOLE_0:def_3; ::_thesis: verum
end;
clusterY \/ X ->  non trivial ;
coherence
 not Y \/ X is trivial ;
end;

registration
let X be non trivial set ;
cluster non trivial finite for Element of bool X;
existence
 ex b1 being Subset of X st
( not b1 is trivial & b1 is finite )
proof
consider x, y being set  such that
A1: x in X and
A2: y in X and
A3: x <> y by ZFMISC_1:def_10;
take  {x,y} ; ::_thesis: ( {x,y} is Subset of X & not {x,y} is trivial & {x,y} is finite )
thus  ( {x,y} is Subset of X & not {x,y} is trivial & {x,y} is finite ) by A1, A2, A3, Th3, ZFMISC_1:32; ::_thesis: verum
end;
end;

theorem Th4: :: CHAIN_1:4
for X, y being set st X is trivial & not X \/ {y} is trivial holds
ex x being set st X = {x}
proof
let X, y be set ; ::_thesis: ( X is trivial & not X \/ {y} is trivial implies ex x being set st X = {x} )
assume that
A1: X is trivial and
A2: not X \/ {y} is trivial ; ::_thesis: ex x being set st X = {x}
 ( X is empty or ex x being set st X = {x} ) by A1, ZFMISC_1:131;
hence  ex x being set st X = {x} by A2; ::_thesis: verum
end;

scheme :: CHAIN_1:sch 2
NonEmptyFinite{ F1() ->  non empty set , F2() ->  non empty finite Subset of F1(), P1[ set ] } :
 P1[F2()]
provided
A1: for x being Element of F1() st x in F2() holds
P1[{x}] and
A2: for x being Element of F1()
for B being non empty finite Subset of F1() st x in F2() & B c= F2() & not x in B & P1[B] holds
P1[B \/ {x}]
proof
defpred S1[ set ] means ( $1 is empty or P1[$1] );
A3: F2() is finite ;
A4: S1[ {} ] ;
A5: for x, B being set st x in F2() & B c= F2() & S1[B] holds
S1[B \/ {x}]
proof
let x, B be set ; ::_thesis: ( x in F2() & B c= F2() & S1[B] implies S1[B \/ {x}] )
assume that
A6: x in F2() and
A7: B c= F2() and
A8: S1[B] ; ::_thesis: S1[B \/ {x}]
reconsider B = B as Subset of F1() by A7, XBOOLE_1:1;
percases ( x in B or ( not B is empty & not x in B ) or B is empty ) ;
suppose x in B ; ::_thesis: S1[B \/ {x}]
then  {x} c= B by ZFMISC_1:31;
hence  S1[B \/ {x}] by A8, XBOOLE_1:12; ::_thesis: verum
end;
suppose ( not B is empty & not x in B ) ; ::_thesis: S1[B \/ {x}]
hence  S1[B \/ {x}] by A2, A6, A7, A8; ::_thesis: verum
end;
suppose B is empty ; ::_thesis: S1[B \/ {x}]
hence  S1[B \/ {x}] by A1, A6; ::_thesis: verum
end;
end;
end;
 S1[F2()] from FINSET_1:sch_2(A3, A4, A5);
hence  P1[F2()] ; ::_thesis: verum
end;

scheme :: CHAIN_1:sch 3
NonTrivialFinite{ F1() ->  non trivial set , F2() ->  non trivial finite Subset of F1(), P1[ set ] } :
 P1[F2()]
provided
A1: for x, y being Element of F1() st x in F2() & y in F2() & x <> y holds
P1[{x,y}] and
A2: for x being Element of F1()
for B being non trivial finite Subset of F1() st x in F2() & B c= F2() & not x in B & P1[B] holds
P1[B \/ {x}]
proof
defpred S1[ set ] means ( $1 is trivial or P1[$1] );
A3: F2() is finite ;
A4: S1[ {} ] ;
A5: for x, B being set st x in F2() & B c= F2() & S1[B] holds
S1[B \/ {x}]
proof
let x, B be set ; ::_thesis: ( x in F2() & B c= F2() & S1[B] implies S1[B \/ {x}] )
assume that
A6: x in F2() and
A7: B c= F2() and
A8: S1[B] ; ::_thesis: S1[B \/ {x}]
reconsider B = B as Subset of F1() by A7, XBOOLE_1:1;
percases ( B \/ {x} is trivial or x in B or ( not B is trivial & not x in B ) or ( B is trivial & not B \/ {x} is trivial ) ) ;
suppose B \/ {x} is trivial ; ::_thesis: S1[B \/ {x}]
hence  S1[B \/ {x}] ; ::_thesis: verum
end;
suppose x in B ; ::_thesis: S1[B \/ {x}]
then  {x} c= B by ZFMISC_1:31;
hence  S1[B \/ {x}] by A8, XBOOLE_1:12; ::_thesis: verum
end;
suppose ( not B is trivial & not x in B ) ; ::_thesis: S1[B \/ {x}]
hence  S1[B \/ {x}] by A2, A6, A7, A8; ::_thesis: verum
end;
supposeA9: ( B is trivial & not B \/ {x} is trivial ) ; ::_thesis: S1[B \/ {x}]
then consider y being set  such that
A10: B = {y} by Th4;
A11: x <> y by A9, A10;
A12: B \/ {x} = {x,y} by A10, ENUMSET1:1;
 y in B by A10, TARSKI:def_1;
hence  S1[B \/ {x}] by A1, A6, A7, A11, A12; ::_thesis: verum
end;
end;
end;
 S1[F2()] from FINSET_1:sch_2(A3, A4, A5);
hence  P1[F2()] ; ::_thesis: verum
end;

theorem Th5: :: CHAIN_1:5
for X being set holds
( card X = 2 iff ex x, y being set st
( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) ) )
proof
let X be set ; ::_thesis: ( card X = 2 iff ex x, y being set st
( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) ) )
hereby  ::_thesis: ( ex x, y being set st
( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) ) implies card X = 2 )
assume A1: card X = 2 ; ::_thesis: ex x, y being set st
( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) )
then reconsider X9 = X as finite set ;
consider x, y being set  such that
A2: x <> y and
A3: X9 = {x,y} by A1, CARD_2:60;
take x = x; ::_thesis: ex y being set st
( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) )
take y = y; ::_thesis: ( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) )
thus  ( x in X & y in X & x <> y & ( for z being set holds
( not z in X or z = x or z = y ) ) ) by A2, A3, TARSKI:def_2; ::_thesis: verum
end;
given x, y being set  such that A4: x in X and
A5: y in X and
A6: x <> y and
A7: for z being set holds
( not z in X or z = x or z = y ) ; ::_thesis: card X = 2
 for z being set holds
( z in X iff ( z = x or z = y ) ) by A4, A5, A7;
then  X = {x,y} by TARSKI:def_2;
hence  card X = 2 by A6, CARD_2:57; ::_thesis: verum
end;

theorem :: CHAIN_1:6
for m, n being Element of NAT holds
( ( m is even iff n is even ) iff m + n is even ) ;

theorem Th7: :: CHAIN_1:7
for X, Y being finite set st X misses Y holds
( ( card X is even iff card Y is even ) iff card (X \/ Y) is even )
proof
let X, Y be finite set ; ::_thesis: ( X misses Y implies ( ( card X is even iff card Y is even ) iff card (X \/ Y) is even ) )
assume  X misses Y ; ::_thesis: ( ( card X is even iff card Y is even ) iff card (X \/ Y) is even )
then  card (X \/ Y) = (card X) + (card Y) by CARD_2:40;
hence  ( ( card X is even iff card Y is even ) iff card (X \/ Y) is even ) ; ::_thesis: verum
end;

theorem Th8: :: CHAIN_1:8
for X, Y being finite set holds
( ( card X is even iff card Y is even ) iff card (X \+\ Y) is even )
proof
let X, Y be finite set ; ::_thesis: ( ( card X is even iff card Y is even ) iff card (X \+\ Y) is even )
A1: X \ Y misses X /\ Y by XBOOLE_1:89;
A2: X = (X \ Y) \/ (X /\ Y) by XBOOLE_1:51;
A3: Y \ X misses X /\ Y by XBOOLE_1:89;
A4: Y = (Y \ X) \/ (X /\ Y) by XBOOLE_1:51;
A5: X \ Y misses Y \ X by XBOOLE_1:82;
A6: X \+\ Y = (X \ Y) \/ (Y \ X) by XBOOLE_0:def_6;
A7: ( ( card (X \ Y) is even iff card (X /\ Y) is even ) iff card X is even ) by A1, A2, Th7;
 ( ( card (Y \ X) is even iff card (X /\ Y) is even ) iff card Y is even ) by A3, A4, Th7;
hence  ( ( card X is even iff card Y is even ) iff card (X \+\ Y) is even ) by A5, A6, A7, Th7; ::_thesis: verum
end;

definition
let n be Element of NAT ;
redefine func REAL n means :Def3: :: CHAIN_1:def 3
 for x being set holds
( x in it iff x is Function of (Seg n),REAL );
compatibility
 for b1 being FinSequenceSet of REAL holds
( b1 = REAL n iff for x being set holds
( x in b1 iff x is Function of (Seg n),REAL ) )
proof
A1: for x being set holds
( x in REAL n iff x is Function of (Seg n),REAL )
proof
let x be set ; ::_thesis: ( x in REAL n iff x is Function of (Seg n),REAL )
hereby  ::_thesis: ( x is Function of (Seg n),REAL implies x in REAL n )
assume  x in REAL n ; ::_thesis: x is Function of (Seg n),REAL
then  x in n -tuples_on REAL by EUCLID:def_1;
then  x in Funcs ((Seg n),REAL) by FINSEQ_2:93;
hence  x is Function of (Seg n),REAL by FUNCT_2:66; ::_thesis: verum
end;
assume  x is Function of (Seg n),REAL ; ::_thesis: x in REAL n
then  x in Funcs ((Seg n),REAL) by FUNCT_2:8;
then  x in n -tuples_on REAL by FINSEQ_2:93;
hence  x in REAL n by EUCLID:def_1; ::_thesis: verum
end;
let X be FinSequenceSet of REAL ; ::_thesis: ( X = REAL n iff for x being set holds
( x in X iff x is Function of (Seg n),REAL ) )
thus  ( X = REAL n implies for x being set holds
( x in X iff x is Function of (Seg n),REAL ) ) by A1; ::_thesis: ( ( for x being set holds
( x in X iff x is Function of (Seg n),REAL ) ) implies X = REAL n )
assume A2: for x being set holds
( x in X iff x is Function of (Seg n),REAL ) ; ::_thesis: X = REAL n
now__::_thesis:_for_x_being_set_holds_
(_x_in_X_iff_x_in_REAL_n_)
let x be set ; ::_thesis: ( x in X iff x in REAL n )
 ( x in X iff x is Function of (Seg n),REAL ) by A2;
hence  ( x in X iff x in REAL n ) by A1; ::_thesis: verum
end;
hence  X = REAL n by TARSKI:1; ::_thesis: verum
end;
end;

:: deftheorem Def3 defines REAL CHAIN_1:def_3_:_
 for n being Element of NAT
for b2 being FinSequenceSet of REAL holds
( b2 = REAL n iff for x being set holds
( x in b2 iff x is Function of (Seg n),REAL ) );

                   begin

definition
let d be non zero Element of NAT ;
mode Grating of d ->  Function of (Seg d),(bool REAL) means :Def4: :: CHAIN_1:def 4
 for i being Element of Seg d holds
( not it . i is trivial & it . i is finite );
existence
 ex b1 being Function of (Seg d),(bool REAL) st
for i being Element of Seg d holds
( not b1 . i is trivial & b1 . i is finite )
proof
defpred S1[ set , set ] means $2 is non trivial finite Subset of REAL;
A1: for i being set st i in Seg d holds
ex X being set st S1[i,X]
proof
let i be set ; ::_thesis: ( i in Seg d implies ex X being set st S1[i,X] )
assume  i in Seg d ; ::_thesis: ex X being set st S1[i,X]
set X = the non trivial finite Subset of REAL;
take  the non trivial finite Subset of REAL ; ::_thesis: S1[i, the non trivial finite Subset of REAL]
thus  S1[i, the non trivial finite Subset of REAL] ; ::_thesis: verum
end;
consider G being Function such that
A2: ( dom G = Seg d & ( for i being set st i in Seg d holds
S1[i,G . i] ) ) from CLASSES1:sch_1(A1);
 for i being set st i in Seg d holds
G . i in bool REAL
proof
let i be set ; ::_thesis: ( i in Seg d implies G . i in bool REAL )
assume  i in Seg d ; ::_thesis: G . i in bool REAL
then  G . i is Subset of REAL by A2;
hence  G . i in bool REAL ; ::_thesis: verum
end;
then reconsider G = G as Function of (Seg d),(bool REAL) by A2, FUNCT_2:3;
take  G ; ::_thesis: for i being Element of Seg d holds
( not G . i is trivial & G . i is finite )
thus  for i being Element of Seg d holds
( not G . i is trivial & G . i is finite ) by A2; ::_thesis: verum
end;
end;

:: deftheorem Def4 defines Grating CHAIN_1:def_4_:_
 for d being non zero Element of NAT
for b2 being Function of (Seg d),(bool REAL) holds
( b2 is Grating of d iff for i being Element of Seg d holds
( not b2 . i is trivial & b2 . i is finite ) );

registration
let d be non zero Element of NAT ;
cluster  -> V25() for Grating of d;
coherence
 for b1 being Grating of d holds b1 is V25()
proof
let G be Grating of d; ::_thesis: G is V25()
let i be set ; :: according to FINSET_1:def_5 ::_thesis: ( not i in Seg d or G . i is finite )
assume  i in Seg d ; ::_thesis: G . i is finite
hence  G . i is finite by Def4; ::_thesis: verum
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let i be Element of Seg d;
:: original: .
redefine funcG . i ->  non trivial finite Subset of REAL;
coherence
 G . i is non trivial finite Subset of REAL by Def4;
end;

theorem Th9: :: CHAIN_1:9
for d being non zero Element of NAT
for x being Element of REAL d
for G being Grating of d holds
( x in product G iff for i being Element of Seg d holds x . i in G . i )
proof
let d be non zero Element of NAT ; ::_thesis: for x being Element of REAL d
for G being Grating of d holds
( x in product G iff for i being Element of Seg d holds x . i in G . i )
let x be Element of REAL d; ::_thesis: for G being Grating of d holds
( x in product G iff for i being Element of Seg d holds x . i in G . i )
let G be Grating of d; ::_thesis: ( x in product G iff for i being Element of Seg d holds x . i in G . i )
 x is Function of (Seg d),REAL by Def3;
then A1: dom x = Seg d by FUNCT_2:def_1;
A2: dom G = Seg d by FUNCT_2:def_1;
hence  ( x in product G implies for i being Element of Seg d holds x . i in G . i ) by CARD_3:9; ::_thesis: ( ( for i being Element of Seg d holds x . i in G . i ) implies x in product G )
assume  for i being Element of Seg d holds x . i in G . i ; ::_thesis: x in product G
then  for i being set st i in Seg d holds
x . i in G . i ;
hence  x in product G by A1, A2, CARD_3:9; ::_thesis: verum
end;

theorem :: CHAIN_1:10
canceled;

theorem Th11: :: CHAIN_1:11
for X being non empty finite Subset of REAL ex ri being Real st
( ri in X & ( for xi being Real st xi in X holds
ri >= xi ) )
proof
defpred S1[ set ] means  ex ri being Real st
( ri in $1 & ( for xi being Real st xi in $1 holds
ri >= xi ) );
let X be non empty finite Subset of REAL; ::_thesis: ex ri being Real st
( ri in X & ( for xi being Real st xi in X holds
ri >= xi ) )
A1: for xi being Real st xi in X holds
S1[{xi}]
proof
let xi be Real; ::_thesis: ( xi in X implies S1[{xi}] )
assume  xi in X ; ::_thesis: S1[{xi}]
take  xi ; ::_thesis: ( xi in {xi} & ( for xi being Real st xi in {xi} holds
xi >= xi ) )
thus  ( xi in {xi} & ( for xi being Real st xi in {xi} holds
xi >= xi ) ) by TARSKI:def_1; ::_thesis: verum
end;
A2: for x being Real
for B being non empty finite Subset of REAL st x in X & B c= X & not x in B & S1[B] holds
S1[B \/ {x}]
proof
let x be Real; ::_thesis: for B being non empty finite Subset of REAL st x in X & B c= X & not x in B & S1[B] holds
S1[B \/ {x}]
let B be non empty finite Subset of REAL; ::_thesis: ( x in X & B c= X & not x in B & S1[B] implies S1[B \/ {x}] )
assume that
 x in X and
 B c= X and
 not x in B and
A3: S1[B] ; ::_thesis: S1[B \/ {x}]
consider ri being Real such that
A4: ri in B and
A5: for xi being Real st xi in B holds
ri >= xi by A3;
set B9 = B \/ {x};
A6: now__::_thesis:_for_xi_being_Real_holds_
(_xi_in_B_\/_{x}_iff_(_xi_in_B_or_xi_=_x_)_)
let xi be Real; ::_thesis: ( xi in B \/ {x} iff ( xi in B or xi = x ) )
 ( xi in {x} iff xi = x ) by TARSKI:def_1;
hence  ( xi in B \/ {x} iff ( xi in B or xi = x ) ) by XBOOLE_0:def_3; ::_thesis: verum
end;
percases ( x <= ri or ri < x ) ;
supposeA7: x <= ri ; ::_thesis: S1[B \/ {x}]
take  ri ; ::_thesis: ( ri in B \/ {x} & ( for xi being Real st xi in B \/ {x} holds
ri >= xi ) )
thus  ri in B \/ {x} by A4, A6; ::_thesis: for xi being Real st xi in B \/ {x} holds
ri >= xi
let xi be Real; ::_thesis: ( xi in B \/ {x} implies ri >= xi )
assume  xi in B \/ {x} ; ::_thesis: ri >= xi
then  ( xi in B or xi = x ) by A6;
hence  ri >= xi by A5, A7; ::_thesis: verum
end;
supposeA8: ri < x ; ::_thesis: S1[B \/ {x}]
take  x ; ::_thesis: ( x in B \/ {x} & ( for xi being Real st xi in B \/ {x} holds
x >= xi ) )
thus  x in B \/ {x} by A6; ::_thesis: for xi being Real st xi in B \/ {x} holds
x >= xi
let xi be Real; ::_thesis: ( xi in B \/ {x} implies x >= xi )
assume  xi in B \/ {x} ; ::_thesis: x >= xi
then  ( xi in B or xi = x ) by A6;
then  ( ri >= xi or xi = x ) by A5;
hence  x >= xi by A8, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  S1[X] from CHAIN_1:sch_2(A1, A2); ::_thesis: verum
end;

theorem Th12: :: CHAIN_1:12
for X being non empty finite Subset of REAL ex li being Real st
( li in X & ( for xi being Real st xi in X holds
li <= xi ) )
proof
defpred S1[ set ] means  ex li being Real st
( li in $1 & ( for xi being Real st xi in $1 holds
li <= xi ) );
let X be non empty finite Subset of REAL; ::_thesis: ex li being Real st
( li in X & ( for xi being Real st xi in X holds
li <= xi ) )
A1: for xi being Real st xi in X holds
S1[{xi}]
proof
let xi be Real; ::_thesis: ( xi in X implies S1[{xi}] )
assume  xi in X ; ::_thesis: S1[{xi}]
take  xi ; ::_thesis: ( xi in {xi} & ( for xi being Real st xi in {xi} holds
xi <= xi ) )
thus  ( xi in {xi} & ( for xi being Real st xi in {xi} holds
xi <= xi ) ) by TARSKI:def_1; ::_thesis: verum
end;
A2: for x being Real
for B being non empty finite Subset of REAL st x in X & B c= X & not x in B & S1[B] holds
S1[B \/ {x}]
proof
let x be Real; ::_thesis: for B being non empty finite Subset of REAL st x in X & B c= X & not x in B & S1[B] holds
S1[B \/ {x}]
let B be non empty finite Subset of REAL; ::_thesis: ( x in X & B c= X & not x in B & S1[B] implies S1[B \/ {x}] )
assume that
 x in X and
 B c= X and
 not x in B and
A3: S1[B] ; ::_thesis: S1[B \/ {x}]
consider li being Real such that
A4: li in B and
A5: for xi being Real st xi in B holds
li <= xi by A3;
set B9 = B \/ {x};
A6: now__::_thesis:_for_xi_being_Real_holds_
(_xi_in_B_\/_{x}_iff_(_xi_in_B_or_xi_=_x_)_)
let xi be Real; ::_thesis: ( xi in B \/ {x} iff ( xi in B or xi = x ) )
 ( xi in {x} iff xi = x ) by TARSKI:def_1;
hence  ( xi in B \/ {x} iff ( xi in B or xi = x ) ) by XBOOLE_0:def_3; ::_thesis: verum
end;
percases ( li <= x or x < li ) ;
supposeA7: li <= x ; ::_thesis: S1[B \/ {x}]
take  li ; ::_thesis: ( li in B \/ {x} & ( for xi being Real st xi in B \/ {x} holds
li <= xi ) )
thus  li in B \/ {x} by A4, A6; ::_thesis: for xi being Real st xi in B \/ {x} holds
li <= xi
let xi be Real; ::_thesis: ( xi in B \/ {x} implies li <= xi )
assume  xi in B \/ {x} ; ::_thesis: li <= xi
then  ( xi in B or xi = x ) by A6;
hence  li <= xi by A5, A7; ::_thesis: verum
end;
supposeA8: x < li ; ::_thesis: S1[B \/ {x}]
take  x ; ::_thesis: ( x in B \/ {x} & ( for xi being Real st xi in B \/ {x} holds
x <= xi ) )
thus  x in B \/ {x} by A6; ::_thesis: for xi being Real st xi in B \/ {x} holds
x <= xi
let xi be Real; ::_thesis: ( xi in B \/ {x} implies x <= xi )
assume  xi in B \/ {x} ; ::_thesis: x <= xi
then  ( xi in B or xi = x ) by A6;
then  ( li <= xi or xi = x ) by A5;
hence  x <= xi by A8, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  S1[X] from CHAIN_1:sch_2(A1, A2); ::_thesis: verum
end;

theorem Th13: :: CHAIN_1:13
for Gi being non trivial finite Subset of REAL ex li, ri being Real st
( li in Gi & ri in Gi & li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) )
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: ex li, ri being Real st
( li in Gi & ri in Gi & li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) )
defpred S1[ set ] means  ex li, ri being Real st
( li in $1 & ri in $1 & li < ri & ( for xi being Real st xi in $1 & li < xi holds
not xi < ri ) );
A1: now__::_thesis:_for_li,_ri_being_Real_st_li_in_Gi_&_ri_in_Gi_&_li_<>_ri_holds_
S1[{li,ri}]
let li, ri be Real; ::_thesis: ( li in Gi & ri in Gi & li <> ri implies S1[{li,ri}] )
assume that
 li in Gi and
 ri in Gi and
A2: li <> ri ; ::_thesis: S1[{li,ri}]
A3: now__::_thesis:_for_li,_ri_being_Real_st_li_<_ri_holds_
S1[{li,ri}]
let li, ri be Real; ::_thesis: ( li < ri implies S1[{li,ri}] )
assume A4: li < ri ; ::_thesis: S1[{li,ri}]
thus  S1[{li,ri}] ::_thesis: verum
proof
take  li ; ::_thesis: ex ri being Real st
( li in {li,ri} & ri in {li,ri} & li < ri & ( for xi being Real st xi in {li,ri} & li < xi holds
not xi < ri ) )
take  ri ; ::_thesis: ( li in {li,ri} & ri in {li,ri} & li < ri & ( for xi being Real st xi in {li,ri} & li < xi holds
not xi < ri ) )
thus  ( li in {li,ri} & ri in {li,ri} & li < ri & ( for xi being Real st xi in {li,ri} & li < xi holds
not xi < ri ) ) by A4, TARSKI:def_2; ::_thesis: verum
end;
end;
 ( li < ri or ri < li ) by A2, XXREAL_0:1;
hence  S1[{li,ri}] by A3; ::_thesis: verum
end;
A5: for x being Real
for B being non trivial finite Subset of REAL st x in Gi & B c= Gi & not x in B & S1[B] holds
S1[B \/ {x}]
proof
let x be Real; ::_thesis: for B being non trivial finite Subset of REAL st x in Gi & B c= Gi & not x in B & S1[B] holds
S1[B \/ {x}]
let B be non trivial finite Subset of REAL; ::_thesis: ( x in Gi & B c= Gi & not x in B & S1[B] implies S1[B \/ {x}] )
assume that
 x in Gi and
 B c= Gi and
A6: not x in B and
A7: S1[B] ; ::_thesis: S1[B \/ {x}]
consider li, ri being Real such that
A8: li in B and
A9: ri in B and
A10: li < ri and
A11: for xi being Real st xi in B & li < xi holds
not xi < ri by A7;
percases ( x < li or ( li < x & x < ri ) or ri < x ) by A6, A8, A9, XXREAL_0:1;
supposeA12: x < li ; ::_thesis: S1[B \/ {x}]
take  li ; ::_thesis: ex ri being Real st
( li in B \/ {x} & ri in B \/ {x} & li < ri & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri ) )
take  ri ; ::_thesis: ( li in B \/ {x} & ri in B \/ {x} & li < ri & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri ) )
thus  ( li in B \/ {x} & ri in B \/ {x} & li < ri ) by A8, A9, A10, XBOOLE_0:def_3; ::_thesis: for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri
let xi be Real; ::_thesis: ( xi in B \/ {x} & li < xi implies not xi < ri )
assume  xi in B \/ {x} ; ::_thesis: ( not li < xi or not xi < ri )
then  ( xi in B or xi in {x} ) by XBOOLE_0:def_3;
hence  ( not li < xi or not xi < ri ) by A11, A12, TARSKI:def_1; ::_thesis: verum
end;
supposeA13: ( li < x & x < ri ) ; ::_thesis: S1[B \/ {x}]
take  li ; ::_thesis: ex ri being Real st
( li in B \/ {x} & ri in B \/ {x} & li < ri & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri ) )
take  x ; ::_thesis: ( li in B \/ {x} & x in B \/ {x} & li < x & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < x ) )
 x in {x} by TARSKI:def_1;
hence  ( li in B \/ {x} & x in B \/ {x} & li < x ) by A8, A13, XBOOLE_0:def_3; ::_thesis: for xi being Real st xi in B \/ {x} & li < xi holds
not xi < x
let xi be Real; ::_thesis: ( xi in B \/ {x} & li < xi implies not xi < x )
assume  xi in B \/ {x} ; ::_thesis: ( not li < xi or not xi < x )
then  ( xi in B or xi in {x} ) by XBOOLE_0:def_3;
then  ( not li < xi or not xi < ri or xi = x ) by A11, TARSKI:def_1;
hence  ( not li < xi or not xi < x ) by A13, XXREAL_0:2; ::_thesis: verum
end;
supposeA14: ri < x ; ::_thesis: S1[B \/ {x}]
take  li ; ::_thesis: ex ri being Real st
( li in B \/ {x} & ri in B \/ {x} & li < ri & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri ) )
take  ri ; ::_thesis: ( li in B \/ {x} & ri in B \/ {x} & li < ri & ( for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri ) )
thus  ( li in B \/ {x} & ri in B \/ {x} & li < ri ) by A8, A9, A10, XBOOLE_0:def_3; ::_thesis: for xi being Real st xi in B \/ {x} & li < xi holds
not xi < ri
let xi be Real; ::_thesis: ( xi in B \/ {x} & li < xi implies not xi < ri )
assume  xi in B \/ {x} ; ::_thesis: ( not li < xi or not xi < ri )
then  ( xi in B or xi in {x} ) by XBOOLE_0:def_3;
hence  ( not li < xi or not xi < ri ) by A11, A14, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
thus  S1[Gi] from CHAIN_1:sch_3(A1, A5); ::_thesis: verum
end;

theorem :: CHAIN_1:14
for X being non empty finite Subset of REAL ex ri being Real st
( ri in X & ( for xi being Real st xi in X holds
ri >= xi ) ) by Th11;

theorem :: CHAIN_1:15
for Gi being non trivial finite Subset of REAL ex li, ri being Real st
( li in Gi & ri in Gi & ri < li & ( for xi being Real st xi in Gi holds
( not xi < ri & not li < xi ) ) )
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: ex li, ri being Real st
( li in Gi & ri in Gi & ri < li & ( for xi being Real st xi in Gi holds
( not xi < ri & not li < xi ) ) )
consider li being Real such that
A1: li in Gi and
A2: for xi being Real st xi in Gi holds
li >= xi by Th11;
consider ri being Real such that
A3: ri in Gi and
A4: for xi being Real st xi in Gi holds
ri <= xi by Th12;
take  li ; ::_thesis: ex ri being Real st
( li in Gi & ri in Gi & ri < li & ( for xi being Real st xi in Gi holds
( not xi < ri & not li < xi ) ) )
take  ri ; ::_thesis: ( li in Gi & ri in Gi & ri < li & ( for xi being Real st xi in Gi holds
( not xi < ri & not li < xi ) ) )
A5: ri <= li by A2, A3;
now__::_thesis:_not_li_=_ri
assume A6: li = ri ; ::_thesis: contradiction
consider x1, x2 being set  such that
A7: x1 in Gi and
A8: x2 in Gi and
A9: x1 <> x2 by ZFMISC_1:def_10;
reconsider x1 = x1, x2 = x2 as Real by A7, A8;
A10: ri <= x1 by A4, A7;
A11: x1 <= li by A2, A7;
A12: ri <= x2 by A4, A8;
A13: x2 <= li by A2, A8;
 x1 = li by A6, A10, A11, XXREAL_0:1;
hence  contradiction by A6, A9, A12, A13, XXREAL_0:1; ::_thesis: verum
end;
hence  ( li in Gi & ri in Gi & ri < li & ( for xi being Real st xi in Gi holds
( not xi < ri & not li < xi ) ) ) by A1, A2, A3, A4, A5, XXREAL_0:1; ::_thesis: verum
end;

definition
let Gi be non trivial finite Subset of REAL;
mode Gap of Gi ->  Element of [:REAL,REAL:] means :Def5: :: CHAIN_1:def 5
 ex li, ri being Real st
( it = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) );
existence
 ex b1 being Element of [:REAL,REAL:] ex li, ri being Real st
( b1 = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
proof
consider li, ri being Real such that
A1: li in Gi and
A2: ri in Gi and
A3: li < ri and
A4: for xi being Real st xi in Gi & li < xi holds
not xi < ri by Th13;
take  [li,ri] ; ::_thesis: ex li, ri being Real st
( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
take  li ; ::_thesis: ex ri being Real st
( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
take  ri ; ::_thesis: ( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
thus  ( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) by A1, A2, A3, A4; ::_thesis: verum
end;
end;

:: deftheorem Def5 defines Gap CHAIN_1:def_5_:_
 for Gi being non trivial finite Subset of REAL
for b2 being Element of [:REAL,REAL:] holds
( b2 is Gap of Gi iff ex li, ri being Real st
( b2 = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) );

theorem Th16: :: CHAIN_1:16
for Gi being non trivial finite Subset of REAL
for li, ri being Real holds
( [li,ri] is Gap of Gi iff ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) )
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for li, ri being Real holds
( [li,ri] is Gap of Gi iff ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) )
let li, ri be Real; ::_thesis: ( [li,ri] is Gap of Gi iff ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) )
thus  ( [li,ri] is Gap of Gi implies ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) ) ::_thesis: ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) implies [li,ri] is Gap of Gi )
proof
assume  [li,ri] is Gap of Gi ; ::_thesis: ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
then consider li9, ri9 being Real such that
A1: [li,ri] = [li9,ri9] and
A2: li9 in Gi and
A3: ri9 in Gi and
A4: ( ( li9 < ri9 & ( for xi being Real st xi in Gi & li9 < xi holds
not xi < ri9 ) ) or ( ri9 < li9 & ( for xi being Real st xi in Gi holds
( not li9 < xi & not xi < ri9 ) ) ) ) by Def5;
A5: li9 = li by A1, XTUPLE_0:1;
 ri9 = ri by A1, XTUPLE_0:1;
hence  ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) ) by A2, A3, A4, A5; ::_thesis: verum
end;
thus  ( li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) implies [li,ri] is Gap of Gi ) by Def5; ::_thesis: verum
end;

theorem :: CHAIN_1:17
for Gi being non trivial finite Subset of REAL
for li, ri, li9, ri9 being Real st Gi = {li,ri} holds
( [li9,ri9] is Gap of Gi iff ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) )
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for li, ri, li9, ri9 being Real st Gi = {li,ri} holds
( [li9,ri9] is Gap of Gi iff ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) )
let li, ri, li9, ri9 be Real; ::_thesis: ( Gi = {li,ri} implies ( [li9,ri9] is Gap of Gi iff ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) ) )
assume A1: Gi = {li,ri} ; ::_thesis: ( [li9,ri9] is Gap of Gi iff ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) )
hereby  ::_thesis: ( ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) implies [li9,ri9] is Gap of Gi )
assume A2: [li9,ri9] is Gap of Gi ; ::_thesis: ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) )
then A3: li9 in Gi by Th16;
A4: ri9 in Gi by A2, Th16;
A5: ( li9 = li or li9 = ri ) by A1, A3, TARSKI:def_2;
 li9 <> ri9 by A2, Th16;
hence  ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) by A1, A4, A5, TARSKI:def_2; ::_thesis: verum
end;
 for Gi being non trivial finite Subset of REAL
for li, ri being Real st Gi = {li,ri} holds
[li,ri] is Gap of Gi
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for li, ri being Real st Gi = {li,ri} holds
[li,ri] is Gap of Gi
let li, ri be Real; ::_thesis: ( Gi = {li,ri} implies [li,ri] is Gap of Gi )
assume A6: Gi = {li,ri} ; ::_thesis: [li,ri] is Gap of Gi
take  li ; :: according to CHAIN_1:def_5 ::_thesis: ex ri being Real st
( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
take  ri ; ::_thesis: ( [li,ri] = [li,ri] & li in Gi & ri in Gi & ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) )
thus  ( [li,ri] = [li,ri] & li in Gi & ri in Gi ) by A6, TARSKI:def_2; ::_thesis: ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) )
 li <> ri by A6, Th3;
hence  ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) by A6, TARSKI:def_2, XXREAL_0:1; ::_thesis: verum
end;
hence  ( ( ( li9 = li & ri9 = ri ) or ( li9 = ri & ri9 = li ) ) implies [li9,ri9] is Gap of Gi ) by A1; ::_thesis: verum
end;

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

theorem Th18: :: CHAIN_1:18
for Gi being non trivial finite Subset of REAL
for xi being Real st xi in Gi holds
ex ri being Real st [xi,ri] is Gap of Gi
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for xi being Real st xi in Gi holds
ex ri being Real st [xi,ri] is Gap of Gi
let xi be Real; ::_thesis: ( xi in Gi implies ex ri being Real st [xi,ri] is Gap of Gi )
assume A1: xi in Gi ; ::_thesis: ex ri being Real st [xi,ri] is Gap of Gi
defpred S1[ Real] means $1 > xi;
set Gi9 =  {  H1(ri9) where ri9 is Real : ( H1(ri9) in Gi & S1[ri9] )  }  ;
A2:  {  H1(ri9) where ri9 is Real : ( H1(ri9) in Gi & S1[ri9] )  }  c= Gi from FRAENKEL:sch_17();
then reconsider Gi9 =  {  H1(ri9) where ri9 is Real : ( H1(ri9) in Gi & S1[ri9] )  }  as finite Subset of REAL by XBOOLE_1:1;
percases ( Gi9 is empty or not Gi9 is empty ) ;
supposeA3: Gi9 is empty ; ::_thesis: ex ri being Real st [xi,ri] is Gap of Gi
A4: now__::_thesis:_for_xi9_being_Real_st_xi9_in_Gi_holds_
not_xi9_>_xi
let xi9 be Real; ::_thesis: ( xi9 in Gi implies not xi9 > xi )
assume that
A5: xi9 in Gi and
A6: xi9 > xi ; ::_thesis: contradiction
 xi9 in Gi9 by A5, A6;
hence  contradiction by A3; ::_thesis: verum
end;
consider li being Real such that
A7: li in Gi and
A8: for xi9 being Real st xi9 in Gi holds
li <= xi9 by Th12;
take  li ; ::_thesis: [xi,li] is Gap of Gi
A9: now__::_thesis:_(_li_=_xi_implies_(_Gi_=_{xi}_&_contradiction_)_)
assume A10: li = xi ; ::_thesis: ( Gi = {xi} & contradiction )
 for xi9 being set holds
( xi9 in Gi iff xi9 = xi )
proof
let xi9 be set ; ::_thesis: ( xi9 in Gi iff xi9 = xi )
hereby  ::_thesis: ( xi9 = xi implies xi9 in Gi )
assume A11: xi9 in Gi ; ::_thesis: xi9 = xi
then reconsider xi99 = xi9 as Element of REAL ;
A12: li <= xi99 by A8, A11;
 xi99 <= xi by A4, A11;
hence  xi9 = xi by A10, A12, XXREAL_0:1; ::_thesis: verum
end;
thus  ( xi9 = xi implies xi9 in Gi ) by A1; ::_thesis: verum
end;
hence  Gi = {xi} by TARSKI:def_1; ::_thesis: contradiction
hence  contradiction ; ::_thesis: verum
end;
 li <= xi by A1, A8;
then A13: li < xi by A9, XXREAL_0:1;
 for xi9 being Real st xi9 in Gi holds
( not xi < xi9 & not xi9 < li ) by A4, A8;
hence  [xi,li] is Gap of Gi by A1, A7, A13, Th16; ::_thesis: verum
end;
suppose not Gi9 is empty ; ::_thesis: ex ri being Real st [xi,ri] is Gap of Gi
then reconsider Gi9 = Gi9 as non empty finite Subset of REAL ;
consider ri being Real such that
A14: ri in Gi9 and
A15: for ri9 being Real st ri9 in Gi9 holds
ri9 >= ri by Th12;
take  ri ; ::_thesis: [xi,ri] is Gap of Gi
now__::_thesis:_(_xi_in_Gi_&_ri_in_Gi_&_xi_<_ri_&_(_for_xi9_being_Real_st_xi9_in_Gi_&_xi_<_xi9_holds_
not_xi9_<_ri_)_)
thus  xi in Gi by A1; ::_thesis: ( ri in Gi & xi < ri & ( for xi9 being Real st xi9 in Gi & xi < xi9 holds
not xi9 < ri ) )
thus  ri in Gi by A2, A14; ::_thesis: ( xi < ri & ( for xi9 being Real st xi9 in Gi & xi < xi9 holds
not xi9 < ri ) )
 ex ri9 being Real st
( ri9 = ri & ri9 in Gi & xi < ri9 ) by A14;
hence  xi < ri ; ::_thesis: for xi9 being Real st xi9 in Gi & xi < xi9 holds
not xi9 < ri
hereby  ::_thesis: verum
let xi9 be Real; ::_thesis: ( xi9 in Gi & xi < xi9 implies not xi9 < ri )
assume  xi9 in Gi ; ::_thesis: ( not xi < xi9 or not xi9 < ri )
then  ( xi9 <= xi or xi9 in Gi9 ) ;
hence  ( not xi < xi9 or not xi9 < ri ) by A15; ::_thesis: verum
end;
end;
hence  [xi,ri] is Gap of Gi by Th16; ::_thesis: verum
end;
end;
end;

theorem Th19: :: CHAIN_1:19
for Gi being non trivial finite Subset of REAL
for xi being Real st xi in Gi holds
ex li being Real st [li,xi] is Gap of Gi
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for xi being Real st xi in Gi holds
ex li being Real st [li,xi] is Gap of Gi
let xi be Real; ::_thesis: ( xi in Gi implies ex li being Real st [li,xi] is Gap of Gi )
assume A1: xi in Gi ; ::_thesis: ex li being Real st [li,xi] is Gap of Gi
defpred S1[ Real] means $1 < xi;
set Gi9 =  {  H1(li9) where li9 is Real : ( H1(li9) in Gi & S1[li9] )  }  ;
A2:  {  H1(li9) where li9 is Real : ( H1(li9) in Gi & S1[li9] )  }  c= Gi from FRAENKEL:sch_17();
then reconsider Gi9 =  {  H1(li9) where li9 is Real : ( H1(li9) in Gi & S1[li9] )  }  as finite Subset of REAL by XBOOLE_1:1;
percases ( Gi9 is empty or not Gi9 is empty ) ;
supposeA3: Gi9 is empty ; ::_thesis: ex li being Real st [li,xi] is Gap of Gi
A4: now__::_thesis:_for_xi9_being_Real_st_xi9_in_Gi_holds_
not_xi9_<_xi
let xi9 be Real; ::_thesis: ( xi9 in Gi implies not xi9 < xi )
assume that
A5: xi9 in Gi and
A6: xi9 < xi ; ::_thesis: contradiction
 xi9 in Gi9 by A5, A6;
hence  contradiction by A3; ::_thesis: verum
end;
consider ri being Real such that
A7: ri in Gi and
A8: for xi9 being Real st xi9 in Gi holds
ri >= xi9 by Th11;
take  ri ; ::_thesis: [ri,xi] is Gap of Gi
A9: now__::_thesis:_(_ri_=_xi_implies_(_Gi_=_{xi}_&_contradiction_)_)
assume A10: ri = xi ; ::_thesis: ( Gi = {xi} & contradiction )
 for xi9 being set holds
( xi9 in Gi iff xi9 = xi )
proof
let xi9 be set ; ::_thesis: ( xi9 in Gi iff xi9 = xi )
hereby  ::_thesis: ( xi9 = xi implies xi9 in Gi )
assume A11: xi9 in Gi ; ::_thesis: xi9 = xi
then reconsider xi99 = xi9 as Element of REAL ;
A12: ri >= xi99 by A8, A11;
 xi99 >= xi by A4, A11;
hence  xi9 = xi by A10, A12, XXREAL_0:1; ::_thesis: verum
end;
thus  ( xi9 = xi implies xi9 in Gi ) by A1; ::_thesis: verum
end;
hence  Gi = {xi} by TARSKI:def_1; ::_thesis: contradiction
hence  contradiction ; ::_thesis: verum
end;
 ri >= xi by A1, A8;
then A13: ri > xi by A9, XXREAL_0:1;
 for xi9 being Real st xi9 in Gi holds
( not xi9 > ri & not xi > xi9 ) by A4, A8;
hence  [ri,xi] is Gap of Gi by A1, A7, A13, Th16; ::_thesis: verum
end;
suppose not Gi9 is empty ; ::_thesis: ex li being Real st [li,xi] is Gap of Gi
then reconsider Gi9 = Gi9 as non empty finite Subset of REAL ;
consider li being Real such that
A14: li in Gi9 and
A15: for li9 being Real st li9 in Gi9 holds
li9 <= li by Th11;
take  li ; ::_thesis: [li,xi] is Gap of Gi
now__::_thesis:_(_xi_in_Gi_&_li_in_Gi_&_xi_>_li_&_(_for_xi9_being_Real_st_xi9_in_Gi_&_xi9_>_li_holds_
not_xi_>_xi9_)_)
thus  xi in Gi by A1; ::_thesis: ( li in Gi & xi > li & ( for xi9 being Real st xi9 in Gi & xi9 > li holds
not xi > xi9 ) )
thus  li in Gi by A2, A14; ::_thesis: ( xi > li & ( for xi9 being Real st xi9 in Gi & xi9 > li holds
not xi > xi9 ) )
 ex li9 being Real st
( li9 = li & li9 in Gi & xi > li9 ) by A14;
hence  xi > li ; ::_thesis: for xi9 being Real st xi9 in Gi & xi9 > li holds
not xi > xi9
hereby  ::_thesis: verum
let xi9 be Real; ::_thesis: ( xi9 in Gi & xi9 > li implies not xi > xi9 )
assume  xi9 in Gi ; ::_thesis: ( not xi9 > li or not xi > xi9 )
then  ( xi9 >= xi or xi9 in Gi9 ) ;
hence  ( not xi9 > li or not xi > xi9 ) by A15; ::_thesis: verum
end;
end;
hence  [li,xi] is Gap of Gi by Th16; ::_thesis: verum
end;
end;
end;

theorem Th20: :: CHAIN_1:20
for Gi being non trivial finite Subset of REAL
for li, ri, ri9 being Real st [li,ri] is Gap of Gi & [li,ri9] is Gap of Gi holds
ri = ri9
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for li, ri, ri9 being Real st [li,ri] is Gap of Gi & [li,ri9] is Gap of Gi holds
ri = ri9
let li, ri, ri9 be Real; ::_thesis: ( [li,ri] is Gap of Gi & [li,ri9] is Gap of Gi implies ri = ri9 )
A1: ( ri <= ri9 & ri9 <= ri implies ri = ri9 ) by XXREAL_0:1;
assume that
A2: [li,ri] is Gap of Gi and
A3: [li,ri9] is Gap of Gi ; ::_thesis: ri = ri9
A4: ri in Gi by A2, Th16;
A5: ri9 in Gi by A3, Th16;
percases ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) by A2, Th16;
supposeA6: ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) ; ::_thesis: ri = ri9
 ( ri9 <= li or ( li < ri9 & ri9 < ri ) or ri <= ri9 ) ;
hence  ri = ri9 by A1, A3, A4, A5, A6, Th16; ::_thesis: verum
end;
supposeA7: ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ; ::_thesis: ri = ri9
 ( ri9 < ri or ( ri <= ri9 & ri9 <= li ) or li < ri9 ) ;
hence  ri = ri9 by A1, A3, A4, A5, A7, Th16; ::_thesis: verum
end;
end;
end;

theorem Th21: :: CHAIN_1:21
for Gi being non trivial finite Subset of REAL
for li, ri, li9 being Real st [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi holds
li = li9
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for li, ri, li9 being Real st [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi holds
li = li9
let li, ri, li9 be Real; ::_thesis: ( [li,ri] is Gap of Gi & [li9,ri] is Gap of Gi implies li = li9 )
A1: ( li <= li9 & li9 <= li implies li = li9 ) by XXREAL_0:1;
assume that
A2: [li,ri] is Gap of Gi and
A3: [li9,ri] is Gap of Gi ; ::_thesis: li = li9
A4: li in Gi by A2, Th16;
A5: li9 in Gi by A3, Th16;
percases ( ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) or ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ) by A2, Th16;
supposeA6: ( li < ri & ( for xi being Real st xi in Gi & li < xi holds
not xi < ri ) ) ; ::_thesis: li = li9
 ( li9 <= li or ( li < li9 & li9 < ri ) or ri <= li9 ) ;
hence  li = li9 by A1, A3, A4, A5, A6, Th16; ::_thesis: verum
end;
supposeA7: ( ri < li & ( for xi being Real st xi in Gi holds
( not li < xi & not xi < ri ) ) ) ; ::_thesis: li = li9
 ( li9 < ri or ( ri <= li9 & li9 <= li ) or li < li9 ) ;
hence  li = li9 by A1, A3, A4, A5, A7, Th16; ::_thesis: verum
end;
end;
end;

theorem Th22: :: CHAIN_1:22
for Gi being non trivial finite Subset of REAL
for ri, li, ri9, li9 being Real st ri < li & [li,ri] is Gap of Gi & ri9 < li9 & [li9,ri9] is Gap of Gi holds
( li = li9 & ri = ri9 )
proof
let Gi be non trivial finite Subset of REAL; ::_thesis: for ri, li, ri9, li9 being Real st ri < li & [li,ri] is Gap of Gi & ri9 < li9 & [li9,ri9] is Gap of Gi holds
( li = li9 & ri = ri9 )
let ri, li, ri9, li9 be Real; ::_thesis: ( ri < li & [li,ri] is Gap of Gi & ri9 < li9 & [li9,ri9] is Gap of Gi implies ( li = li9 & ri = ri9 ) )
assume that
A1: ri < li and
A2: [li,ri] is Gap of Gi and
A3: ri9 < li9 and
A4: [li9,ri9] is Gap of Gi ; ::_thesis: ( li = li9 & ri = ri9 )
A5: li in Gi by A2, Th16;
A6: ri in Gi by A2, Th16;
A7: li9 in Gi by A4, Th16;
A8: ri9 in Gi by A4, Th16;
hereby  ::_thesis: ri = ri9
assume  li <> li9 ; ::_thesis: contradiction
then  ( li < li9 or li9 < li ) by XXREAL_0:1;
hence  contradiction by A1, A2, A3, A4, A5, A7, Th16; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume  ri <> ri9 ; ::_thesis: contradiction
then  ( ri < ri9 or ri9 < ri ) by XXREAL_0:1;
hence  contradiction by A1, A2, A3, A4, A6, A8, Th16; ::_thesis: verum
end;
end;

definition
let d be non zero Element of NAT ;
let l, r be Element of REAL d;
func cell (l,r) ->  non empty Subset of (REAL d) equals :: CHAIN_1:def 6
 {  x where x is Element of REAL d : ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )  }  ;
coherence
  {  x where x is Element of REAL d : ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )  }  is non empty Subset of (REAL d)
proof
defpred S1[ Element of REAL d] means ( for i being Element of Seg d holds
( l . i <= $1 . i & $1 . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( $1 . i <= r . i or l . i <= $1 . i ) ) );
set CELL =  {  x where x is Element of REAL d : S1[x]  }  ;
 S1[l] ;
then A1: l in  {  x where x is Element of REAL d : S1[x]  }  ;
  {  x where x is Element of REAL d : S1[x]  }  c= REAL d from FRAENKEL:sch_10();
hence   {  x where x is Element of REAL d : ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )  }  is non empty Subset of (REAL d) by A1; ::_thesis: verum
end;
end;

:: deftheorem  defines cell CHAIN_1:def_6_:_
 for d being non zero Element of NAT
for l, r being Element of REAL d holds cell (l,r) =  {  x where x is Element of REAL d : ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )  }  ;

theorem Th23: :: CHAIN_1:23
for d being non zero Element of NAT
for x, l, r being Element of REAL d holds
( x in cell (l,r) iff ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for x, l, r being Element of REAL d holds
( x in cell (l,r) iff ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) )
let x, l, r be Element of REAL d; ::_thesis: ( x in cell (l,r) iff ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) )
defpred S1[ Element of REAL d] means ( for i being Element of Seg d holds
( l . i <= $1 . i & $1 . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( $1 . i <= r . i or l . i <= $1 . i ) ) );
A1: cell (l,r) =  {  x9 where x9 is Element of REAL d : S1[x9]  }  ;
thus  ( x in cell (l,r) iff S1[x] ) from LMOD_7:sch_7(A1); ::_thesis: verum
end;

theorem Th24: :: CHAIN_1:24
for d being non zero Element of NAT
for l, r, x being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i ) holds
( x in cell (l,r) iff for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r, x being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i ) holds
( x in cell (l,r) iff for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) )
let l, r, x be Element of REAL d; ::_thesis: ( ( for i being Element of Seg d holds l . i <= r . i ) implies ( x in cell (l,r) iff for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) ) )
assume A1: for i being Element of Seg d holds l . i <= r . i ; ::_thesis: ( x in cell (l,r) iff for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) )
hereby  ::_thesis: ( ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) ) implies x in cell (l,r) )
assume  x in cell (l,r) ; ::_thesis: for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i )
then  ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) by Th23;
hence  for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) by A1; ::_thesis: verum
end;
thus  ( ( for i being Element of Seg d holds
( l . i <= x . i & x . i <= r . i ) ) implies x in cell (l,r) ) ; ::_thesis: verum
end;

theorem Th25: :: CHAIN_1:25
for d being non zero Element of NAT
for r, l, x being Element of REAL d st ex i being Element of Seg d st r . i < l . i holds
( x in cell (l,r) iff ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for r, l, x being Element of REAL d st ex i being Element of Seg d st r . i < l . i holds
( x in cell (l,r) iff ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )
let r, l, x be Element of REAL d; ::_thesis: ( ex i being Element of Seg d st r . i < l . i implies ( x in cell (l,r) iff ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) )
given i0 being Element of Seg d such that A1: r . i0 < l . i0 ; ::_thesis: ( x in cell (l,r) iff ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) )
 ( x . i0 < l . i0 or r . i0 < x . i0 ) by A1, XXREAL_0:2;
hence  ( x in cell (l,r) implies ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) ) by Th23; ::_thesis: ( ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) implies x in cell (l,r) )
thus  ( ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) ) implies x in cell (l,r) ) ; ::_thesis: verum
end;

theorem Th26: :: CHAIN_1:26
for d being non zero Element of NAT
for l, r being Element of REAL d holds
( l in cell (l,r) & r in cell (l,r) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d holds
( l in cell (l,r) & r in cell (l,r) )
let l, r be Element of REAL d; ::_thesis: ( l in cell (l,r) & r in cell (l,r) )
A1: ( for i being Element of Seg d holds
( l . i <= l . i & l . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( l . i <= r . i or l . i <= l . i ) ) ) ;
 ( for i being Element of Seg d holds
( l . i <= r . i & r . i <= r . i ) or ex i being Element of Seg d st
( r . i < l . i & ( r . i <= r . i or l . i <= r . i ) ) ) ;
hence  ( l in cell (l,r) & r in cell (l,r) ) by A1; ::_thesis: verum
end;

theorem Th27: :: CHAIN_1:27
for d being non zero Element of NAT
for x being Element of REAL d holds cell (x,x) = {x}
proof
let d be non zero Element of NAT ; ::_thesis: for x being Element of REAL d holds cell (x,x) = {x}
let x be Element of REAL d; ::_thesis: cell (x,x) = {x}
 for x9 being set holds
( x9 in cell (x,x) iff x9 = x )
proof
let x9 be set ; ::_thesis: ( x9 in cell (x,x) iff x9 = x )
thus  ( x9 in cell (x,x) implies x9 = x ) ::_thesis: ( x9 = x implies x9 in cell (x,x) )
proof
assume A1: x9 in cell (x,x) ; ::_thesis: x9 = x
then reconsider x = x, x9 = x9 as Function of (Seg d),REAL by Def3;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_x9_._i_=_x_._i
let i be Element of Seg d; ::_thesis: x9 . i = x . i
A2: for i being Element of Seg d holds x . i <= x . i ;
then A3: x . i <= x9 . i by A1, Th24;
 x9 . i <= x . i by A1, A2, Th24;
hence  x9 . i = x . i by A3, XXREAL_0:1; ::_thesis: verum
end;
hence  x9 = x by FUNCT_2:63; ::_thesis: verum
end;
thus  ( x9 = x implies x9 in cell (x,x) ) by Th26; ::_thesis: verum
end;
hence  cell (x,x) = {x} by TARSKI:def_1; ::_thesis: verum
end;

theorem Th28: :: CHAIN_1:28
for d being non zero Element of NAT
for l9, r9, l, r being Element of REAL d st ( for i being Element of Seg d holds l9 . i <= r9 . i ) holds
( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l9, r9, l, r being Element of REAL d st ( for i being Element of Seg d holds l9 . i <= r9 . i ) holds
( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) )
let l9, r9, l, r be Element of REAL d; ::_thesis: ( ( for i being Element of Seg d holds l9 . i <= r9 . i ) implies ( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) ) )
assume A1: for i being Element of Seg d holds l9 . i <= r9 . i ; ::_thesis: ( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) )
thus  ( cell (l,r) c= cell (l9,r9) implies for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) ) ::_thesis: ( ( for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) ) implies cell (l,r) c= cell (l9,r9) )
proof
assume A2: cell (l,r) c= cell (l9,r9) ; ::_thesis: for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i )
let i0 be Element of Seg d; ::_thesis: ( l9 . i0 <= l . i0 & l . i0 <= r . i0 & r . i0 <= r9 . i0 )
percases ( r . i0 < l . i0 or l . i0 <= r . i0 ) ;
supposeA3: r . i0 < l . i0 ; ::_thesis: ( l9 . i0 <= l . i0 & l . i0 <= r . i0 & r . i0 <= r9 . i0 )
defpred S1[ Element of Seg d, Real] means ( $2 > l . $1 & $2 > r9 . $1 );
A4: for i being Element of Seg d ex xi being Real st S1[i,xi] by Th2;
consider x being Function of (Seg d),REAL such that
A5: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A4);
reconsider x = x as Element of REAL d by Def3;
 ex i being Element of Seg d st
( r . i < l . i & ( x . i <= r . i or l . i <= x . i ) )
proof
take  i0 ; ::_thesis: ( r . i0 < l . i0 & ( x . i0 <= r . i0 or l . i0 <= x . i0 ) )
thus  ( r . i0 < l . i0 & ( x . i0 <= r . i0 or l . i0 <= x . i0 ) ) by A3, A5; ::_thesis: verum
end;
then A6: x in cell (l,r) ;
 ex i being Element of Seg d st
( x . i < l9 . i or r9 . i < x . i )
proof
take  i0 ; ::_thesis: ( x . i0 < l9 . i0 or r9 . i0 < x . i0 )
thus  ( x . i0 < l9 . i0 or r9 . i0 < x . i0 ) by A5; ::_thesis: verum
end;
hence  ( l9 . i0 <= l . i0 & l . i0 <= r . i0 & r . i0 <= r9 . i0 ) by A1, A2, A6, Th24; ::_thesis: verum
end;
supposeA7: l . i0 <= r . i0 ; ::_thesis: ( l9 . i0 <= l . i0 & l . i0 <= r . i0 & r . i0 <= r9 . i0 )
A8: l in cell (l,r) by Th26;
 r in cell (l,r) by Th26;
hence  ( l9 . i0 <= l . i0 & l . i0 <= r . i0 & r . i0 <= r9 . i0 ) by A1, A2, A7, A8, Th24; ::_thesis: verum
end;
end;
end;
assume A9: for i being Element of Seg d holds
( l9 . i <= l . i & l . i <= r . i & r . i <= r9 . i ) ; ::_thesis: cell (l,r) c= cell (l9,r9)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in cell (l,r) or x in cell (l9,r9) )
assume A10: x in cell (l,r) ; ::_thesis: x in cell (l9,r9)
then reconsider x = x as Element of REAL d ;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l9_._i_<=_x_._i_&_x_._i_<=_r9_._i_&_l9_._i_<=_r9_._i_)
let i be Element of Seg d; ::_thesis: ( l9 . i <= x . i & x . i <= r9 . i & l9 . i <= r9 . i )
A11: l9 . i <= l . i by A9;
A12: l . i <= x . i by A9, A10, Th24;
A13: x . i <= r . i by A9, A10, Th24;
 r . i <= r9 . i by A9;
hence  ( l9 . i <= x . i & x . i <= r9 . i ) by A11, A12, A13, XXREAL_0:2; ::_thesis: l9 . i <= r9 . i
hence  l9 . i <= r9 . i by XXREAL_0:2; ::_thesis: verum
end;
hence  x in cell (l9,r9) ; ::_thesis: verum
end;

theorem Th29: :: CHAIN_1:29
for d being non zero Element of NAT
for r, l, l9, r9 being Element of REAL d st ( for i being Element of Seg d holds r . i < l . i ) holds
( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) )
proof
let d be non zero Element of NAT ; ::_thesis: for r, l, l9, r9 being Element of REAL d st ( for i being Element of Seg d holds r . i < l . i ) holds
( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) )
let r, l, l9, r9 be Element of REAL d; ::_thesis: ( ( for i being Element of Seg d holds r . i < l . i ) implies ( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) ) )
assume A1: for i being Element of Seg d holds r . i < l . i ; ::_thesis: ( cell (l,r) c= cell (l9,r9) iff for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) )
thus  ( cell (l,r) c= cell (l9,r9) implies for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) ) ::_thesis: ( ( for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) ) implies cell (l,r) c= cell (l9,r9) )
proof
assume A2: cell (l,r) c= cell (l9,r9) ; ::_thesis: for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i )
A3: for i being Element of Seg d holds r9 . i < l9 . i
proof
let i0 be Element of Seg d; ::_thesis: r9 . i0 < l9 . i0
assume A4: l9 . i0 <= r9 . i0 ; ::_thesis: contradiction
defpred S1[ Element of Seg d, Real] means ( ( $1 = i0 implies ( l . $1 < $2 & r9 . $1 < $2 ) ) & ( r9 . $1 < l9 . $1 implies ( r9 . $1 < $2 & $2 < l9 . $1 ) ) );
A5: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
percases ( ( i = i0 & r9 . i < l9 . i ) or i <> i0 or l9 . i <= r9 . i ) ;
suppose ( i = i0 & r9 . i < l9 . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
hence  ex xi being Real st S1[i,xi] by A4; ::_thesis: verum
end;
supposeA6: i <> i0 ; ::_thesis: ex xi being Real st S1[i,xi]
 ( r9 . i < l9 . i implies ex xi being Real st
( r9 . i < xi & xi < l9 . i ) ) by Th1;
hence  ex xi being Real st S1[i,xi] by A6; ::_thesis: verum
end;
supposeA7: l9 . i <= r9 . i ; ::_thesis: ex xi being Real st S1[i,xi]
 ex xi being Real st
( l . i < xi & r9 . i < xi ) by Th2;
hence  ex xi being Real st S1[i,xi] by A7; ::_thesis: verum
end;
end;
end;
consider x being Function of (Seg d),REAL such that
A8: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A5);
reconsider x = x as Element of REAL d by Def3;
A9: r . i0 < l . i0 by A1;
 ( x . i0 <= r . i0 or l . i0 <= x . i0 ) by A8;
then A10: x in cell (l,r) by A9;
percases ( for i being Element of Seg d holds
( l9 . i <= x . i & x . i <= r9 . i ) or ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) ) by A2, A10, Th23;
suppose for i being Element of Seg d holds
( l9 . i <= x . i & x . i <= r9 . i ) ; ::_thesis: contradiction
then  x . i0 <= r9 . i0 ;
hence  contradiction by A8; ::_thesis: verum
end;
suppose ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) ; ::_thesis: contradiction
hence  contradiction by A8; ::_thesis: verum
end;
end;
end;
let i0 be Element of Seg d; ::_thesis: ( r . i0 <= r9 . i0 & r9 . i0 < l9 . i0 & l9 . i0 <= l . i0 )
hereby  ::_thesis: ( r9 . i0 < l9 . i0 & l9 . i0 <= l . i0 )
assume A11: r9 . i0 < r . i0 ; ::_thesis: contradiction
defpred S1[ Element of Seg d, Real] means ( r9 . $1 < $2 & $2 < l9 . $1 & ( $1 = i0 implies $2 < r . $1 ) );
A12: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
percases ( ( i = i0 & l9 . i <= r . i ) or ( i = i0 & r . i <= l9 . i ) or i <> i0 ) ;
supposeA13: ( i = i0 & l9 . i <= r . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
 r9 . i < l9 . i by A3;
then consider xi being Real such that
A14: r9 . i < xi and
A15: xi < l9 . i by Th1;
 xi < r . i by A13, A15, XXREAL_0:2;
hence  ex xi being Real st S1[i,xi] by A14, A15; ::_thesis: verum
end;
supposeA16: ( i = i0 & r . i <= l9 . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
then consider xi being Real such that
A17: r9 . i < xi and
A18: xi < r . i by A11, Th1;
 xi < l9 . i by A16, A18, XXREAL_0:2;
hence  ex xi being Real st S1[i,xi] by A17, A18; ::_thesis: verum
end;
supposeA19: i <> i0 ; ::_thesis: ex xi being Real st S1[i,xi]
 r9 . i < l9 . i by A3;
then  ex xi being Real st
( r9 . i < xi & xi < l9 . i ) by Th1;
hence  ex xi being Real st S1[i,xi] by A19; ::_thesis: verum
end;
end;
end;
consider x being Function of (Seg d),REAL such that
A20: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A12);
reconsider x = x as Element of REAL d by Def3;
A21: r . i0 < l . i0 by A1;
 ( x . i0 <= r . i0 or l . i0 <= x . i0 ) by A20;
then A22: x in cell (l,r) by A21;
 ( not l9 . i0 <= x . i0 or not x . i0 <= r9 . i0 ) by A3, XXREAL_0:2;
then  ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) by A2, A22, Th23;
hence  contradiction by A20; ::_thesis: verum
end;
thus  r9 . i0 < l9 . i0 by A3; ::_thesis: l9 . i0 <= l . i0
hereby  ::_thesis: verum
assume A23: l9 . i0 > l . i0 ; ::_thesis: contradiction
defpred S1[ Element of Seg d, Real] means ( l9 . $1 > $2 & $2 > r9 . $1 & ( $1 = i0 implies $2 > l . $1 ) );
A24: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
percases ( ( i = i0 & r9 . i >= l . i ) or ( i = i0 & l . i >= r9 . i ) or i <> i0 ) ;
supposeA25: ( i = i0 & r9 . i >= l . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
 l9 . i > r9 . i by A3;
then consider xi being Real such that
A26: r9 . i < xi and
A27: xi < l9 . i by Th1;
 xi > l . i by A25, A26, XXREAL_0:2;
hence  ex xi being Real st S1[i,xi] by A26, A27; ::_thesis: verum
end;
supposeA28: ( i = i0 & l . i >= r9 . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
then consider xi being Real such that
A29: l . i < xi and
A30: xi < l9 . i by A23, Th1;
 xi > r9 . i by A28, A29, XXREAL_0:2;
hence  ex xi being Real st S1[i,xi] by A29, A30; ::_thesis: verum
end;
supposeA31: i <> i0 ; ::_thesis: ex xi being Real st S1[i,xi]
 l9 . i > r9 . i by A3;
then  ex xi being Real st
( r9 . i < xi & xi < l9 . i ) by Th1;
hence  ex xi being Real st S1[i,xi] by A31; ::_thesis: verum
end;
end;
end;
consider x being Function of (Seg d),REAL such that
A32: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A24);
reconsider x = x as Element of REAL d by Def3;
A33: l . i0 > r . i0 by A1;
 ( x . i0 >= l . i0 or r . i0 >= x . i0 ) by A32;
then A34: x in cell (l,r) by A33;
 ( not r9 . i0 >= x . i0 or not x . i0 >= l9 . i0 ) by A3, XXREAL_0:2;
then  ex i being Element of Seg d st
( l9 . i > r9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) by A2, A34, Th23;
hence  contradiction by A32; ::_thesis: verum
end;
end;
assume A35: for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) ; ::_thesis: cell (l,r) c= cell (l9,r9)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in cell (l,r) or x in cell (l9,r9) )
assume A36: x in cell (l,r) ; ::_thesis: x in cell (l9,r9)
then reconsider x = x as Element of REAL d ;
set i0 = the Element of Seg d;
A37: r . the Element of Seg d <= r9 . the Element of Seg d by A35;
 r9 . the Element of Seg d < l9 . the Element of Seg d by A35;
then A38: r . the Element of Seg d < l9 . the Element of Seg d by A37, XXREAL_0:2;
 l9 . the Element of Seg d <= l . the Element of Seg d by A35;
then  r . the Element of Seg d < l . the Element of Seg d by A38, XXREAL_0:2;
then  ( x . the Element of Seg d < l . the Element of Seg d or r . the Element of Seg d < x . the Element of Seg d ) by XXREAL_0:2;
then consider i being Element of Seg d such that
 r . i < l . i and
A39: ( x . i <= r . i or l . i <= x . i ) by A36, Th23;
A40: r . i <= r9 . i by A35;
A41: l9 . i <= l . i by A35;
A42: r9 . i < l9 . i by A35;
 ( x . i <= r9 . i or l9 . i <= x . i ) by A39, A40, A41, XXREAL_0:2;
hence  x in cell (l9,r9) by A42; ::_thesis: verum
end;

theorem Th30: :: CHAIN_1:30
for d being non zero Element of NAT
for l, r, r9, l9 being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i ) & ( for i being Element of Seg d holds r9 . i < l9 . i ) holds
( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r, r9, l9 being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i ) & ( for i being Element of Seg d holds r9 . i < l9 . i ) holds
( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) )
let l, r, r9, l9 be Element of REAL d; ::_thesis: ( ( for i being Element of Seg d holds l . i <= r . i ) & ( for i being Element of Seg d holds r9 . i < l9 . i ) implies ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) )
assume A1: for i being Element of Seg d holds l . i <= r . i ; ::_thesis: ( ex i being Element of Seg d st not r9 . i < l9 . i or ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) )
assume A2: for i being Element of Seg d holds r9 . i < l9 . i ; ::_thesis: ( cell (l,r) c= cell (l9,r9) iff ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) )
thus  ( cell (l,r) c= cell (l9,r9) implies ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) ) ::_thesis: ( ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i ) implies cell (l,r) c= cell (l9,r9) )
proof
assume A3: cell (l,r) c= cell (l9,r9) ; ::_thesis: ex i being Element of Seg d st
( r . i <= r9 . i or l9 . i <= l . i )
assume A4: for i being Element of Seg d holds
( r9 . i < r . i & l . i < l9 . i ) ; ::_thesis: contradiction
defpred S1[ Element of Seg d, Real] means ( l . $1 <= $2 & $2 <= r . $1 & r9 . $1 < $2 & $2 < l9 . $1 );
A5: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
percases ( ( l . i <= r9 . i & l9 . i <= r . i ) or ( r9 . i < l . i & l9 . i <= r . i ) or r . i < l9 . i ) ;
supposeA6: ( l . i <= r9 . i & l9 . i <= r . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
 r9 . i < l9 . i by A2;
then consider xi being Real such that
A7: r9 . i < xi and
A8: xi < l9 . i by Th1;
take  xi ; ::_thesis: S1[i,xi]
thus  S1[i,xi] by A6, A7, A8, XXREAL_0:2; ::_thesis: verum
end;
supposeA9: ( r9 . i < l . i & l9 . i <= r . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
take  l . i ; ::_thesis: S1[i,l . i]
thus  S1[i,l . i] by A1, A4, A9; ::_thesis: verum
end;
supposeA10: r . i < l9 . i ; ::_thesis: ex xi being Real st S1[i,xi]
take  r . i ; ::_thesis: S1[i,r . i]
thus  S1[i,r . i] by A1, A4, A10; ::_thesis: verum
end;
end;
end;
consider x being Function of (Seg d),REAL such that
A11: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A5);
reconsider x = x as Element of REAL d by Def3;
A12: x in cell (l,r) by A11;
set i0 = the Element of Seg d;
 r9 . the Element of Seg d < l9 . the Element of Seg d by A2;
then  ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) ) by A3, A12, Th25;
hence  contradiction by A11; ::_thesis: verum
end;
given i0 being Element of Seg d such that A13: ( r . i0 <= r9 . i0 or l9 . i0 <= l . i0 ) ; ::_thesis: cell (l,r) c= cell (l9,r9)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in cell (l,r) or x in cell (l9,r9) )
assume A14: x in cell (l,r) ; ::_thesis: x in cell (l9,r9)
then reconsider x = x as Element of REAL d ;
A15: l . i0 <= x . i0 by A1, A14, Th24;
A16: x . i0 <= r . i0 by A1, A14, Th24;
 ex i being Element of Seg d st
( r9 . i < l9 . i & ( x . i <= r9 . i or l9 . i <= x . i ) )
proof
take  i0 ; ::_thesis: ( r9 . i0 < l9 . i0 & ( x . i0 <= r9 . i0 or l9 . i0 <= x . i0 ) )
thus  ( r9 . i0 < l9 . i0 & ( x . i0 <= r9 . i0 or l9 . i0 <= x . i0 ) ) by A2, A13, A15, A16, XXREAL_0:2; ::_thesis: verum
end;
hence  x in cell (l9,r9) ; ::_thesis: verum
end;

theorem Th31: :: CHAIN_1:31
for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i or for i being Element of Seg d holds l . i > r . i ) holds
( cell (l,r) = cell (l9,r9) iff ( l = l9 & r = r9 ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r, l9, r9 being Element of REAL d st ( for i being Element of Seg d holds l . i <= r . i or for i being Element of Seg d holds l . i > r . i ) holds
( cell (l,r) = cell (l9,r9) iff ( l = l9 & r = r9 ) )
let l, r, l9, r9 be Element of REAL d; ::_thesis: ( ( for i being Element of Seg d holds l . i <= r . i or for i being Element of Seg d holds l . i > r . i ) implies ( cell (l,r) = cell (l9,r9) iff ( l = l9 & r = r9 ) ) )
assume A1: ( for i being Element of Seg d holds l . i <= r . i or for i being Element of Seg d holds l . i > r . i ) ; ::_thesis: ( cell (l,r) = cell (l9,r9) iff ( l = l9 & r = r9 ) )
thus  ( cell (l,r) = cell (l9,r9) implies ( l = l9 & r = r9 ) ) ::_thesis: ( l = l9 & r = r9 implies cell (l,r) = cell (l9,r9) )
proof
assume A2: cell (l,r) = cell (l9,r9) ; ::_thesis: ( l = l9 & r = r9 )
percases ( for i being Element of Seg d holds l . i <= r . i or for i being Element of Seg d holds l . i > r . i ) by A1;
supposeA3: for i being Element of Seg d holds l . i <= r . i ; ::_thesis: ( l = l9 & r = r9 )
then A4: for i being Element of Seg d holds
( l . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r . i ) by A2, Th28;
reconsider l = l, r = r, l9 = l9, r9 = r9 as Function of (Seg d),REAL by Def3;
A5: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l_._i_=_l9_._i
let i be Element of Seg d; ::_thesis: l . i = l9 . i
A6: l . i <= l9 . i by A2, A3, Th28;
 l9 . i <= l . i by A2, A4, Th28;
hence  l . i = l9 . i by A6, XXREAL_0:1; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_r_._i_=_r9_._i
let i be Element of Seg d; ::_thesis: r . i = r9 . i
A7: r . i <= r9 . i by A2, A4, Th28;
 r9 . i <= r . i by A2, A3, Th28;
hence  r . i = r9 . i by A7, XXREAL_0:1; ::_thesis: verum
end;
hence  ( l = l9 & r = r9 ) by A5, FUNCT_2:63; ::_thesis: verum
end;
supposeA8: for i being Element of Seg d holds l . i > r . i ; ::_thesis: ( l = l9 & r = r9 )
then A9: for i being Element of Seg d holds
( r . i <= r9 . i & r9 . i < l9 . i & l9 . i <= l . i ) by A2, Th29;
reconsider l = l, r = r, l9 = l9, r9 = r9 as Function of (Seg d),REAL by Def3;
A10: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l_._i_=_l9_._i
let i be Element of Seg d; ::_thesis: l . i = l9 . i
A11: l . i <= l9 . i by A2, A9, Th29;
 l9 . i <= l . i by A2, A8, Th29;
hence  l . i = l9 . i by A11, XXREAL_0:1; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_r_._i_=_r9_._i
let i be Element of Seg d; ::_thesis: r . i = r9 . i
A12: r . i <= r9 . i by A2, A8, Th29;
 r9 . i <= r . i by A2, A9, Th29;
hence  r . i = r9 . i by A12, XXREAL_0:1; ::_thesis: verum
end;
hence  ( l = l9 & r = r9 ) by A10, FUNCT_2:63; ::_thesis: verum
end;
end;
end;
thus  ( l = l9 & r = r9 implies cell (l,r) = cell (l9,r9) ) ; ::_thesis: verum
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
assume A1: k <= d ;
func cells (k,G) ->  non empty finite Subset-Family of (REAL d) equals :Def7: :: CHAIN_1:def 7
 {  (cell (l,r)) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )  }  ;
coherence
  {  (cell (l,r)) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )  }  is non empty finite Subset-Family of (REAL d)
proof
defpred S1[ Element of REAL d, Element of REAL d] means ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & $1 . i < $2 . i & [($1 . i),($2 . i)] is Gap of G . i ) or ( not i in X & $1 . i = $2 . i & $1 . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( $2 . i < $1 . i & [($1 . i),($2 . i)] is Gap of G . i ) ) ) );
deffunc H2( Element of REAL d, Element of REAL d) ->  non empty Subset of (REAL d) = cell ($1,$2);
set CELLS =  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  ;
reconsider X = Seg k as Subset of (Seg d) by A1, FINSEQ_1:5;
defpred S2[ Element of Seg d, Element of [:REAL,REAL:]] means ( ( $1 in X & ex li, ri being Real st
( $2 = [li,ri] & li < ri & $2 is Gap of G . $1 ) ) or ( not $1 in X & ex li being Real st
( $2 = [li,li] & li in G . $1 ) ) );
A2: now__::_thesis:_for_i_being_Element_of_Seg_d_ex_lri_being_Element_of_[:REAL,REAL:]_st_S2[i,lri]
let i be Element of Seg d; ::_thesis: ex lri being Element of [:REAL,REAL:] st S2[i,lri]
thus  ex lri being Element of [:REAL,REAL:] st S2[i,lri] ::_thesis: verum
proof
percases ( i in X or not i in X ) ;
supposeA3: i in X ; ::_thesis: ex lri being Element of [:REAL,REAL:] st S2[i,lri]
consider li, ri being Real such that
A4: li in G . i and
A5: ri in G . i and
A6: li < ri and
A7: for xi being Real st xi in G . i & li < xi holds
not xi < ri by Th13;
take  [li,ri] ; ::_thesis: S2[i,[li,ri]]
 [li,ri] is Gap of G . i by A4, A5, A6, A7, Def5;
hence  S2[i,[li,ri]] by A3, A6; ::_thesis: verum
end;
supposeA8: not i in X ; ::_thesis: ex lri being Element of [:REAL,REAL:] st S2[i,lri]
set li = the Element of G . i;
reconsider li = the Element of G . i as Real ;
reconsider lri = [li,li] as Element of [:REAL,REAL:] ;
take  lri ; ::_thesis: S2[i,lri]
thus  S2[i,lri] by A8; ::_thesis: verum
end;
end;
end;
end;
consider lr being Function of (Seg d),[:REAL,REAL:] such that
A9: for i being Element of Seg d holds S2[i,lr . i] from FUNCT_2:sch_3(A2);
deffunc H3( Element of Seg d) ->  Element of REAL = (lr . $1) `1 ;
consider l being Function of (Seg d),REAL such that
A10: for i being Element of Seg d holds l . i = H3(i) from FUNCT_2:sch_4();
deffunc H4( Element of Seg d) ->  Element of REAL = (lr . $1) `2 ;
consider r being Function of (Seg d),REAL such that
A11: for i being Element of Seg d holds r . i = H4(i) from FUNCT_2:sch_4();
reconsider l = l, r = r as Element of REAL d by Def3;
A12: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_lr_._i_=_[(l_._i),(r_._i)]
let i be Element of Seg d; ::_thesis: lr . i = [(l . i),(r . i)]
A13: l . i = (lr . i) `1 by A10;
 r . i = (lr . i) `2 by A11;
hence  lr . i = [(l . i),(r . i)] by A13, MCART_1:21; ::_thesis: verum
end;
now__::_thesis:_ex_A_being_non_empty_Subset_of_(REAL_d)_st_
(_A_=_cell_(l,r)_&_ex_l,_r_being_Element_of_REAL_d_st_
(_A_=_cell_(l,r)_&_(_ex_X_being_Subset_of_(Seg_d)_st_
(_card_X_=_k_&_(_for_i_being_Element_of_Seg_d_holds_
(_(_i_in_X_&_l_._i_<_r_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)_or_(_not_i_in_X_&_l_._i_=_r_._i_&_l_._i_in_G_._i_)_)_)_)_or_(_k_=_d_&_(_for_i_being_Element_of_Seg_d_holds_
(_r_._i_<_l_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)_)_)_)_)_)
take A = cell (l,r); ::_thesis: ( A = cell (l,r) & ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
thus  A = cell (l,r) ; ::_thesis: ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
now__::_thesis:_ex_X_being_Subset_of_(Seg_d)_st_
(_card_X_=_k_&_ex_l,_r_being_Element_of_REAL_d_st_
(_A_=_cell_(l,r)_&_(_for_i_being_Element_of_Seg_d_holds_
(_(_i_in_X_&_l_._i_<_r_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)_or_(_not_i_in_X_&_l_._i_=_r_._i_&_l_._i_in_G_._i_)_)_)_)_)
take X = X; ::_thesis: ( card X = k & ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( ( b7 in b4 & b5 . b7 < b6 . b7 & [(b5 . b7),(b6 . b7)] is Gap of G . b7 ) or ( not b7 in b4 & b5 . b7 = b6 . b7 & b5 . b7 in G . b7 ) ) ) ) )
thus  card X = k by FINSEQ_1:57; ::_thesis: ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( ( b7 in b4 & b5 . b7 < b6 . b7 & [(b5 . b7),(b6 . b7)] is Gap of G . b7 ) or ( not b7 in b4 & b5 . b7 = b6 . b7 & b5 . b7 in G . b7 ) ) ) )
take l = l; ::_thesis: ex r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( ( b6 in b3 & b4 . b6 < b5 . b6 & [(b4 . b6),(b5 . b6)] is Gap of G . b6 ) or ( not b6 in b3 & b4 . b6 = b5 . b6 & b4 . b6 in G . b6 ) ) ) )
take r = r; ::_thesis: ( A = cell (l,r) & ( for i being Element of Seg d holds
( ( b5 in b2 & b3 . b5 < b4 . b5 & [(b3 . b5),(b4 . b5)] is Gap of G . b5 ) or ( not b5 in b2 & b3 . b5 = b4 . b5 & b3 . b5 in G . b5 ) ) ) )
thus  A = cell (l,r) ; ::_thesis: for i being Element of Seg d holds
( ( b5 in b2 & b3 . b5 < b4 . b5 & [(b3 . b5),(b4 . b5)] is Gap of G . b5 ) or ( not b5 in b2 & b3 . b5 = b4 . b5 & b3 . b5 in G . b5 ) )
let i be Element of Seg d; ::_thesis: ( ( b4 in b1 & b2 . b4 < b3 . b4 & [(b2 . b4),(b3 . b4)] is Gap of G . b4 ) or ( not b4 in b1 & b2 . b4 = b3 . b4 & b2 . b4 in G . b4 ) )
percases ( i in X or not i in X ) ;
supposeA14: i in X ; ::_thesis: ( ( b4 in b1 & b2 . b4 < b3 . b4 & [(b2 . b4),(b3 . b4)] is Gap of G . b4 ) or ( not b4 in b1 & b2 . b4 = b3 . b4 & b2 . b4 in G . b4 ) )
then consider li, ri being Real such that
A15: lr . i = [li,ri] and
A16: li < ri and
A17: lr . i is Gap of G . i by A9;
A18: lr . i = [(l . i),(r . i)] by A12;
then  li = l . i by A15, XTUPLE_0:1;
hence  ( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A14, A15, A16, A17, A18, XTUPLE_0:1; ::_thesis: verum
end;
supposeA19: not i in X ; ::_thesis: ( ( b4 in b1 & b2 . b4 < b3 . b4 & [(b2 . b4),(b3 . b4)] is Gap of G . b4 ) or ( not b4 in b1 & b2 . b4 = b3 . b4 & b2 . b4 in G . b4 ) )
then consider li being Real such that
A20: lr . i = [li,li] and
A21: li in G . i by A9;
A22: [li,li] = [(l . i),(r . i)] by A12, A20;
then  li = l . i by XTUPLE_0:1;
hence  ( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A19, A21, A22, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
hence  ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) ; ::_thesis: verum
end;
then A23: cell (l,r) in  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  ;
defpred S3[ set , Element of REAL d, Element of REAL d, set ] means ( $2 in product G & $3 in product G & ( ( $4 = [0,[$2,$3]] & $1 = cell ($2,$3) ) or ( $4 = [1,[$2,$3]] & $1 = cell ($2,$3) ) ) );
defpred S4[ set , set ] means  ex l, r being Element of REAL d st S3[$1,l,r,$2];
A24: for A being set st A in  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  holds
ex lr being set st S4[A,lr]
proof
let A be set ; ::_thesis: ( A in  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  implies ex lr being set st S4[A,lr] )
assume  A in  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  ; ::_thesis: ex lr being set st S4[A,lr]
then consider l, r being Element of REAL d such that
A25: A = cell (l,r) and
A26: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ;
percases ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A26;
supposeA27: ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) ; ::_thesis: ex lr being set st S4[A,lr]
take  [0,[l,r]] ; ::_thesis: S4[A,[0,[l,r]]]
take  l ; ::_thesis: ex r being Element of REAL d st S3[A,l,r,[0,[l,r]]]
take  r ; ::_thesis: S3[A,l,r,[0,[l,r]]]
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l_._i_in_G_._i_&_r_._i_in_G_._i_)
let i be Element of Seg d; ::_thesis: ( l . i in G . i & r . i in G . i )
 ( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) by A27;
hence  ( l . i in G . i & r . i in G . i ) by Th16; ::_thesis: verum
end;
hence  ( l in product G & r in product G ) by Th9; ::_thesis: ( ( [0,[l,r]] = [0,[l,r]] & A = cell (l,r) ) or ( [0,[l,r]] = [1,[l,r]] & A = cell (l,r) ) )
thus  ( ( [0,[l,r]] = [0,[l,r]] & A = cell (l,r) ) or ( [0,[l,r]] = [1,[l,r]] & A = cell (l,r) ) ) by A25; ::_thesis: verum
end;
supposeA28: ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ; ::_thesis: ex lr being set st S4[A,lr]
take  [1,[l,r]] ; ::_thesis: S4[A,[1,[l,r]]]
take  l ; ::_thesis: ex r being Element of REAL d st S3[A,l,r,[1,[l,r]]]
take  r ; ::_thesis: S3[A,l,r,[1,[l,r]]]
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l_._i_in_G_._i_&_r_._i_in_G_._i_)
let i be Element of Seg d; ::_thesis: ( l . i in G . i & r . i in G . i )
 [(l . i),(r . i)] is Gap of G . i by A28;
hence  ( l . i in G . i & r . i in G . i ) by Th16; ::_thesis: verum
end;
hence  ( l in product G & r in product G ) by Th9; ::_thesis: ( ( [1,[l,r]] = [0,[l,r]] & A = cell (l,r) ) or ( [1,[l,r]] = [1,[l,r]] & A = cell (l,r) ) )
thus  ( ( [1,[l,r]] = [0,[l,r]] & A = cell (l,r) ) or ( [1,[l,r]] = [1,[l,r]] & A = cell (l,r) ) ) by A25; ::_thesis: verum
end;
end;
end;
consider f being Function such that
A29: ( dom f =  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  & ( for A being set st A in  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  holds
S4[A,f . A] ) ) from CLASSES1:sch_1(A24);
A30: f is one-to-one
proof
let A, A9 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not A in dom f or not A9 in dom f or not f . A = f . A9 or A = A9 )
assume that
A31: A in dom f and
A32: A9 in dom f and
A33: f . A = f . A9 ; ::_thesis: A = A9
consider l, r being Element of REAL d such that
A34: S3[A,l,r,f . A] by A29, A31;
consider l9, r9 being Element of REAL d such that
A35: S3[A9,l9,r9,f . A9] by A29, A32;
percases ( ( f . A = [0,[l,r]] & A = cell (l,r) ) or ( f . A = [1,[l,r]] & A = cell (l,r) ) ) by A34;
supposeA36: ( f . A = [0,[l,r]] & A = cell (l,r) ) ; ::_thesis: A = A9
then A37: [l,r] = [l9,r9] by A33, A35, XTUPLE_0:1;
then  l = l9 by XTUPLE_0:1;
hence  A = A9 by A35, A36, A37, XTUPLE_0:1; ::_thesis: verum
end;
supposeA38: ( f . A = [1,[l,r]] & A = cell (l,r) ) ; ::_thesis: A = A9
then A39: [l,r] = [l9,r9] by A33, A35, XTUPLE_0:1;
then  l = l9 by XTUPLE_0:1;
hence  A = A9 by A35, A38, A39, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
reconsider X = product G as finite set ;
A40: rng f c= [:{0,1},[:X,X:]:]
proof
let lr be set ; :: according to TARSKI:def_3 ::_thesis: ( not lr in rng f or lr in [:{0,1},[:X,X:]:] )
assume  lr in rng f ; ::_thesis: lr in [:{0,1},[:X,X:]:]
then consider A being set  such that
A41: A in dom f and
A42: lr = f . A by FUNCT_1:def_3;
consider l, r being Element of REAL d such that
A43: S3[A,l,r,f . A] by A29, A41;
A44: 0 in {0,1} by TARSKI:def_2;
A45: 1 in {0,1} by TARSKI:def_2;
 [l,r] in [:X,X:] by A43, ZFMISC_1:87;
hence  lr in [:{0,1},[:X,X:]:] by A42, A43, A44, A45, ZFMISC_1:87; ::_thesis: verum
end;
  {  H2(l,r) where l, r is Element of REAL d : S1[l,r]  }  c= bool (REAL d) from CHAIN_1:sch_1();
hence   {  (cell (l,r)) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )  }  is non empty finite Subset-Family of (REAL d) by A23, A29, A30, A40, CARD_1:59; ::_thesis: verum
end;
end;

:: deftheorem Def7 defines cells CHAIN_1:def_7_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT st k <= d holds
cells (k,G) =  {  (cell (l,r)) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )  }  ;

theorem Th32: :: CHAIN_1:32
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st k <= d holds
for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st k <= d holds
for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st k <= d holds
for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
let G be Grating of d; ::_thesis: ( k <= d implies for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) ) )
assume  k <= d ; ::_thesis: for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
then  cells (k,G) =  {  (cell (l,r)) where l, r is Element of REAL d : ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )  }  by Def7;
hence  for A being Subset of (REAL d) holds
( A in cells (k,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) ) ; ::_thesis: verum
end;

theorem Th33: :: CHAIN_1:33
for k being Element of NAT
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d st k <= d holds
( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
let G be Grating of d; ::_thesis: ( k <= d implies ( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) ) )
assume A1: k <= d ; ::_thesis: ( cell (l,r) in cells (k,G) iff ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
hereby  ::_thesis: ( ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) implies cell (l,r) in cells (k,G) )
assume  cell (l,r) in cells (k,G) ; ::_thesis: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
then consider l9, r9 being Element of REAL d such that
A2: cell (l,r) = cell (l9,r9) and
A3: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) ) ) by A1, Th32;
 ( l = l9 & r = r9 )
proof
percases ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) by A3;
suppose ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) ; ::_thesis: ( l = l9 & r = r9 )
then  for i being Element of Seg d holds l9 . i <= r9 . i ;
hence  ( l = l9 & r = r9 ) by A2, Th31; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ; ::_thesis: ( l = l9 & r = r9 )
hence  ( l = l9 & r = r9 ) by A2, Th31; ::_thesis: verum
end;
end;
end;
hence  ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A3; ::_thesis: verum
end;
thus  ( ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) implies cell (l,r) in cells (k,G) ) by A1, Th32; ::_thesis: verum
end;

theorem Th34: :: CHAIN_1:34
for k being Element of NAT
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) & ex i being Element of Seg d st
( not ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( l . i = r . i & l . i in G . i ) ) holds
for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) & ex i being Element of Seg d st
( not ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( l . i = r . i & l . i in G . i ) ) holds
for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) & ex i being Element of Seg d st
( not ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( l . i = r . i & l . i in G . i ) ) holds
for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d st k <= d & cell (l,r) in cells (k,G) & ex i being Element of Seg d st
( not ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( l . i = r . i & l . i in G . i ) ) holds
for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
let G be Grating of d; ::_thesis: ( k <= d & cell (l,r) in cells (k,G) & ex i being Element of Seg d st
( not ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( l . i = r . i & l . i in G . i ) ) implies for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
assume that
A1: k <= d and
A2: cell (l,r) in cells (k,G) ; ::_thesis: ( for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
percases ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A1, A2, Th33;
suppose ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) ; ::_thesis: ( for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
hence  ( for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ; ::_thesis: verum
end;
suppose ( k = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ; ::_thesis: ( for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
hence  ( for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ; ::_thesis: verum
end;
end;
end;

theorem Th35: :: CHAIN_1:35
for k being Element of NAT
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) holds
for i being Element of Seg d holds
( l . i in G . i & r . i in G . i )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) holds
for i being Element of Seg d holds
( l . i in G . i & r . i in G . i )
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) holds
for i being Element of Seg d holds
( l . i in G . i & r . i in G . i )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d st k <= d & cell (l,r) in cells (k,G) holds
for i being Element of Seg d holds
( l . i in G . i & r . i in G . i )
let G be Grating of d; ::_thesis: ( k <= d & cell (l,r) in cells (k,G) implies for i being Element of Seg d holds
( l . i in G . i & r . i in G . i ) )
assume that
A1: k <= d and
A2: cell (l,r) in cells (k,G) ; ::_thesis: for i being Element of Seg d holds
( l . i in G . i & r . i in G . i )
let i be Element of Seg d; ::_thesis: ( l . i in G . i & r . i in G . i )
 ( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) or ( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) by A1, A2, Th34;
hence  ( l . i in G . i & r . i in G . i ) by Th16; ::_thesis: verum
end;

theorem :: CHAIN_1:36
for k being Element of NAT
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st k <= d & cell (l,r) in cells (k,G) & not  for i being Element of Seg d holds l . i <= r . i holds
for i being Element of Seg d holds r . i < l . i by Th34;

theorem Th37: :: CHAIN_1:37
for d being non zero Element of NAT
for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (0,G) iff ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (0,G) iff ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) )
let G be Grating of d; ::_thesis: for A being Subset of (REAL d) holds
( A in cells (0,G) iff ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) )
let A be Subset of (REAL d); ::_thesis: ( A in cells (0,G) iff ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) )
hereby  ::_thesis: ( ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) implies A in cells (0,G) )
assume  A in cells (0,G) ; ::_thesis: ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) )
then consider l, r being Element of REAL d such that
A1: A = cell (l,r) and
A2: ( ex X being Subset of (Seg d) st
( card X = 0 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( 0 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by Th32;
consider X being Subset of (Seg d) such that
A3: card X = 0 and
A4: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A2;
reconsider l9 = l, r9 = r as Function of (Seg d),REAL by Def3;
 X = {} by A3;
then A5: for i being Element of Seg d holds
( l9 . i = r9 . i & l . i in G . i ) by A4;
then  l9 = r9 by FUNCT_2:63;
hence  ex x being Element of REAL d st
( A = cell (x,x) & ( for i being Element of Seg d holds x . i in G . i ) ) by A1, A5; ::_thesis: verum
end;
given x being Element of REAL d such that A6: A = cell (x,x) and
A7: for i being Element of Seg d holds x . i in G . i ; ::_thesis: A in cells (0,G)
 ex X being Subset of (Seg d) st
( card X = 0 & ( for i being Element of Seg d holds
( ( i in X & x . i < x . i & [(x . i),(x . i)] is Gap of G . i ) or ( not i in X & x . i = x . i & x . i in G . i ) ) ) )
proof
reconsider X = {} as Subset of (Seg d) by XBOOLE_1:2;
take  X ; ::_thesis: ( card X = 0 & ( for i being Element of Seg d holds
( ( i in X & x . i < x . i & [(x . i),(x . i)] is Gap of G . i ) or ( not i in X & x . i = x . i & x . i in G . i ) ) ) )
thus  ( card X = 0 & ( for i being Element of Seg d holds
( ( i in X & x . i < x . i & [(x . i),(x . i)] is Gap of G . i ) or ( not i in X & x . i = x . i & x . i in G . i ) ) ) ) by A7; ::_thesis: verum
end;
hence  A in cells (0,G) by A6, Th32; ::_thesis: verum
end;

theorem Th38: :: CHAIN_1:38
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d holds
( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
let G be Grating of d; ::_thesis: ( cell (l,r) in cells (0,G) iff ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) )
hereby  ::_thesis: ( l = r & ( for i being Element of Seg d holds l . i in G . i ) implies cell (l,r) in cells (0,G) )
assume  cell (l,r) in cells (0,G) ; ::_thesis: ( l = r & ( for i being Element of Seg d holds l . i in G . i ) )
then consider x being Element of REAL d such that
A1: cell (l,r) = cell (x,x) and
A2: for i being Element of Seg d holds x . i in G . i by Th37;
A3: for i being Element of Seg d holds x . i <= x . i ;
then  l = x by A1, Th31;
hence  ( l = r & ( for i being Element of Seg d holds l . i in G . i ) ) by A1, A2, A3, Th31; ::_thesis: verum
end;
thus  ( l = r & ( for i being Element of Seg d holds l . i in G . i ) implies cell (l,r) in cells (0,G) ) by Th37; ::_thesis: verum
end;

theorem Th39: :: CHAIN_1:39
for d being non zero Element of NAT
for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (d,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (d,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
let G be Grating of d; ::_thesis: for A being Subset of (REAL d) holds
( A in cells (d,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
let A be Subset of (REAL d); ::_thesis: ( A in cells (d,G) iff ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
hereby  ::_thesis: ( ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) implies A in cells (d,G) )
assume  A in cells (d,G) ; ::_thesis: ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
then consider l, r being Element of REAL d such that
A1: A = cell (l,r) and
A2: ( ex X being Subset of (Seg d) st
( card X = d & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( d = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by Th32;
thus  ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) ::_thesis: verum
proof
take  l ; ::_thesis: ex r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
take  r ; ::_thesis: ( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
percases ( ex X being Subset of (Seg d) st
( card X = d & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) by A2;
suppose ex X being Subset of (Seg d) st
( card X = d & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) ; ::_thesis: ( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
then consider X being Subset of (Seg d) such that
A3: card X = d and
A4: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ;
 card X = card (Seg d) by A3, FINSEQ_1:57;
then  not X c< Seg d by CARD_2:48;
then  X = Seg d by XBOOLE_0:def_8;
hence  ( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) by A1, A4; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ; ::_thesis: ( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
hence  ( A = cell (l,r) & ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) by A1; ::_thesis: verum
end;
end;
end;
end;
given l, r being Element of REAL d such that A5: A = cell (l,r) and
A6: for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i and
A7: ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ; ::_thesis: A in cells (d,G)
percases ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) by A7;
supposeA8: for i being Element of Seg d holds l . i < r . i ; ::_thesis: A in cells (d,G)
 ex X being Subset of (Seg d) st
( card X = d & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) )
proof
 Seg d c= Seg d ;
then reconsider X = Seg d as Subset of (Seg d) ;
take  X ; ::_thesis: ( card X = d & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) )
thus  card X = d by FINSEQ_1:57; ::_thesis: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) )
thus  for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A6, A8; ::_thesis: verum
end;
hence  A in cells (d,G) by A5, Th32; ::_thesis: verum
end;
suppose for i being Element of Seg d holds r . i < l . i ; ::_thesis: A in cells (d,G)
hence  A in cells (d,G) by A5, A6, Th32; ::_thesis: verum
end;
end;
end;

theorem Th40: :: CHAIN_1:40
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d holds
( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
let G be Grating of d; ::_thesis: ( cell (l,r) in cells (d,G) iff ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) )
hereby  ::_thesis: ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) in cells (d,G) )
assume  cell (l,r) in cells (d,G) ; ::_thesis: ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) )
then consider l9, r9 being Element of REAL d such that
A1: cell (l,r) = cell (l9,r9) and
A2: for i being Element of Seg d holds [(l9 . i),(r9 . i)] is Gap of G . i and
A3: ( for i being Element of Seg d holds l9 . i < r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by Th39;
A4: ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by A3;
then A5: l = l9 by A1, Th31;
 r = r9 by A1, A4, Th31;
hence  ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) ) by A2, A3, A5; ::_thesis: verum
end;
thus  ( ( for i being Element of Seg d holds [(l . i),(r . i)] is Gap of G . i ) & ( for i being Element of Seg d holds l . i < r . i or for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) in cells (d,G) ) by Th39; ::_thesis: verum
end;

theorem Th41: :: CHAIN_1:41
for d9 being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st d = d9 + 1 holds
for A being Subset of (REAL d) holds
( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
proof
let d9 be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st d = d9 + 1 holds
for A being Subset of (REAL d) holds
( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st d = d9 + 1 holds
for A being Subset of (REAL d) holds
( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
let G be Grating of d; ::_thesis: ( d = d9 + 1 implies for A being Subset of (REAL d) holds
( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
assume A1: d = d9 + 1 ; ::_thesis: for A being Subset of (REAL d) holds
( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
then A2: d9 < d by NAT_1:13;
let A be Subset of (REAL d); ::_thesis: ( A in cells (d9,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
hereby  ::_thesis: ( ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) implies A in cells (d9,G) )
assume  A in cells (d9,G) ; ::_thesis: ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
then consider l, r being Element of REAL d such that
A3: A = cell (l,r) and
A4: ( ex X being Subset of (Seg d) st
( card X = d9 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( d9 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A2, Th32;
take l = l; ::_thesis: ex r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
take r = r; ::_thesis: ex i0 being Element of Seg d st
( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
consider X being Subset of (Seg d) such that
A5: card X = d9 and
A6: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A1, A4;
card ((Seg d) \ X) = (card (Seg d)) - (card X) by CARD_2:44
.= d - d9 by A5, FINSEQ_1:57
.= 1 by A1 ;
then consider i0 being set  such that
A7: (Seg d) \ X = {i0} by CARD_2:42;
 i0 in (Seg d) \ X by A7, TARSKI:def_1;
then reconsider i0 = i0 as Element of Seg d by XBOOLE_0:def_5;
take i0 = i0; ::_thesis: ( A = cell (l,r) & l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
A8: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_i_in_X_iff_i_<>_i0_)
let i be Element of Seg d; ::_thesis: ( i in X iff i <> i0 )
 ( i in (Seg d) \ X iff i = i0 ) by A7, TARSKI:def_1;
hence  ( i in X iff i <> i0 ) by XBOOLE_0:def_5; ::_thesis: verum
end;
thus  A = cell (l,r) by A3; ::_thesis: ( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
 not i0 in X by A8;
hence  ( l . i0 = r . i0 & l . i0 in G . i0 ) by A6; ::_thesis: for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i )
let i be Element of Seg d; ::_thesis: ( i <> i0 implies ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) )
assume  i <> i0 ; ::_thesis: ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i )
then  i in X by A8;
hence  ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) by A6; ::_thesis: verum
end;
given l, r being Element of REAL d, i0 being Element of Seg d such that A9: A = cell (l,r) and
A10: l . i0 = r . i0 and
A11: l . i0 in G . i0 and
A12: for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ; ::_thesis: A in cells (d9,G)
reconsider X = (Seg d) \ {i0} as Subset of (Seg d) by XBOOLE_1:36;
 ( card X = d9 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) )
proof
thus  card X = (card (Seg d)) - (card {i0}) by CARD_2:44
.= d - (card {i0}) by FINSEQ_1:57
.= d - 1 by CARD_1:30
.= d9 by A1 ; ::_thesis: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) )
let i be Element of Seg d; ::_thesis: ( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) )
 ( i in {i0} iff i = i0 ) by TARSKI:def_1;
hence  ( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A10, A11, A12, XBOOLE_0:def_5; ::_thesis: verum
end;
hence  A in cells (d9,G) by A2, A9, Th32; ::_thesis: verum
end;

theorem :: CHAIN_1:42
for d9 being Element of NAT
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st d = d9 + 1 holds
( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
proof
let d9 be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st d = d9 + 1 holds
( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d st d = d9 + 1 holds
( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d st d = d9 + 1 holds
( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
let G be Grating of d; ::_thesis: ( d = d9 + 1 implies ( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) )
assume A1: d = d9 + 1 ; ::_thesis: ( cell (l,r) in cells (d9,G) iff ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) )
hereby  ::_thesis: ( ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) implies cell (l,r) in cells (d9,G) )
assume  cell (l,r) in cells (d9,G) ; ::_thesis: ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
then consider l9, r9 being Element of REAL d, i0 being Element of Seg d such that
A2: cell (l,r) = cell (l9,r9) and
A3: l9 . i0 = r9 . i0 and
A4: l9 . i0 in G . i0 and
A5: for i being Element of Seg d st i <> i0 holds
( l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) by A1, Th41;
take i0 = i0; ::_thesis: ( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) )
A6: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l9_._i_<=_r9_._i
let i be Element of Seg d; ::_thesis: l9 . i <= r9 . i
 ( i = i0 or i <> i0 ) ;
hence  l9 . i <= r9 . i by A3, A5; ::_thesis: verum
end;
then A7: l = l9 by A2, Th31;
 r = r9 by A2, A6, Th31;
hence  ( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) by A3, A4, A5, A7; ::_thesis: verum
end;
thus  ( ex i0 being Element of Seg d st
( l . i0 = r . i0 & l . i0 in G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) ) ) implies cell (l,r) in cells (d9,G) ) by A1, Th41; ::_thesis: verum
end;

theorem Th43: :: CHAIN_1:43
for d being non zero Element of NAT
for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (1,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for A being Subset of (REAL d) holds
( A in cells (1,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
let G be Grating of d; ::_thesis: for A being Subset of (REAL d) holds
( A in cells (1,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
A1: d >= 1 by Def2;
let A be Subset of (REAL d); ::_thesis: ( A in cells (1,G) iff ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
hereby  ::_thesis: ( ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) implies A in cells (1,G) )
assume  A in cells (1,G) ; ::_thesis: ex l, r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
then consider l, r being Element of REAL d such that
A2: A = cell (l,r) and
A3: ( ex X being Subset of (Seg d) st
( card X = 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( 1 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A1, Th32;
take l = l; ::_thesis: ex r being Element of REAL d ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
take r = r; ::_thesis: ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
thus  ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) ::_thesis: verum
proof
percases ( ex X being Subset of (Seg d) st
( card X = 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( d = 1 & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A3;
suppose ex X being Subset of (Seg d) st
( card X = 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) ; ::_thesis: ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
then consider X being Subset of (Seg d) such that
A4: card X = 1 and
A5: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ;
consider i0 being set  such that
A6: X = {i0} by A4, CARD_2:42;
A7: i0 in X by A6, TARSKI:def_1;
then reconsider i0 = i0 as Element of Seg d ;
take  i0 ; ::_thesis: ( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
thus  ( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 ) by A2, A5, A7; ::_thesis: for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i )
let i be Element of Seg d; ::_thesis: ( i <> i0 implies ( l . i = r . i & l . i in G . i ) )
 ( not i in X iff i <> i0 ) by A6, TARSKI:def_1;
hence  ( i <> i0 implies ( l . i = r . i & l . i in G . i ) ) by A5; ::_thesis: verum
end;
supposeA8: ( d = 1 & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ; ::_thesis: ex i0 being Element of Seg d st
( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
reconsider i0 = 1 as Element of Seg d by A1, FINSEQ_1:1;
take  i0 ; ::_thesis: ( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
thus  ( A = cell (l,r) & ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 ) by A2, A8; ::_thesis: for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i )
let i be Element of Seg d; ::_thesis: ( i <> i0 implies ( l . i = r . i & l . i in G . i ) )
A9: 1 <= i by FINSEQ_1:1;
 i <= d by FINSEQ_1:1;
hence  ( i <> i0 implies ( l . i = r . i & l . i in G . i ) ) by A8, A9, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
given l, r being Element of REAL d, i0 being Element of Seg d such that A10: A = cell (l,r) and
A11: ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) and
A12: [(l . i0),(r . i0)] is Gap of G . i0 and
A13: for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ; ::_thesis: A in cells (1,G)
set X = {i0};
percases ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) by A11;
supposeA14: l . i0 < r . i0 ; ::_thesis: A in cells (1,G)
A15: card {i0} = 1 by CARD_1:30;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_(_i_in_{i0}_&_l_._i_<_r_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)_or_(_not_i_in_{i0}_&_l_._i_=_r_._i_&_l_._i_in_G_._i_)_)
let i be Element of Seg d; ::_thesis: ( ( i in {i0} & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in {i0} & l . i = r . i & l . i in G . i ) )
 ( i in {i0} iff i = i0 ) by TARSKI:def_1;
hence  ( ( i in {i0} & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in {i0} & l . i = r . i & l . i in G . i ) ) by A12, A13, A14; ::_thesis: verum
end;
hence  A in cells (1,G) by A1, A10, A15, Th32; ::_thesis: verum
end;
supposeA16: ( d = 1 & r . i0 < l . i0 ) ; ::_thesis: A in cells (1,G)
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_r_._i_<_l_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)
let i be Element of Seg d; ::_thesis: ( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
A17: 1 <= i by FINSEQ_1:1;
A18: i <= d by FINSEQ_1:1;
A19: 1 <= i0 by FINSEQ_1:1;
A20: i0 <= d by FINSEQ_1:1;
A21: i = 1 by A16, A17, A18, XXREAL_0:1;
 i0 = 1 by A16, A19, A20, XXREAL_0:1;
hence  ( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A12, A16, A21; ::_thesis: verum
end;
hence  A in cells (1,G) by A10, A11, Th32; ::_thesis: verum
end;
end;
end;

theorem :: CHAIN_1:44
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (1,G) iff ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) in cells (1,G) iff ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d holds
( cell (l,r) in cells (1,G) iff ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
let G be Grating of d; ::_thesis: ( cell (l,r) in cells (1,G) iff ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) )
hereby  ::_thesis: ( ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) implies cell (l,r) in cells (1,G) )
assume  cell (l,r) in cells (1,G) ; ::_thesis: ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) )
then consider l9, r9 being Element of REAL d, i0 being Element of Seg d such that
A1: cell (l,r) = cell (l9,r9) and
A2: ( l9 . i0 < r9 . i0 or ( d = 1 & r9 . i0 < l9 . i0 ) ) and
A3: [(l9 . i0),(r9 . i0)] is Gap of G . i0 and
A4: for i being Element of Seg d st i <> i0 holds
( l9 . i = r9 . i & l9 . i in G . i ) by Th43;
A5: ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i )
proof
percases ( l9 . i0 < r9 . i0 or ( d = 1 & r9 . i0 < l9 . i0 ) ) by A2;
supposeA6: l9 . i0 < r9 . i0 ; ::_thesis: ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i )
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l9_._i_<=_r9_._i
let i be Element of Seg d; ::_thesis: l9 . i <= r9 . i
 ( i = i0 or i <> i0 ) ;
hence  l9 . i <= r9 . i by A4, A6; ::_thesis: verum
end;
hence  ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) ; ::_thesis: verum
end;
supposeA7: ( d = 1 & r9 . i0 < l9 . i0 ) ; ::_thesis: ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i )
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_r9_._i_<_l9_._i
let i be Element of Seg d; ::_thesis: r9 . i < l9 . i
A8: 1 <= i by FINSEQ_1:1;
A9: i <= d by FINSEQ_1:1;
A10: 1 <= i0 by FINSEQ_1:1;
A11: i0 <= d by FINSEQ_1:1;
A12: i = 1 by A7, A8, A9, XXREAL_0:1;
 i0 = 1 by A7, A10, A11, XXREAL_0:1;
hence  r9 . i < l9 . i by A7, A12; ::_thesis: verum
end;
hence  ( for i being Element of Seg d holds l9 . i <= r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) ; ::_thesis: verum
end;
end;
end;
then A13: l = l9 by A1, Th31;
 r = r9 by A1, A5, Th31;
hence  ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) by A2, A3, A4, A13; ::_thesis: verum
end;
thus  ( ex i0 being Element of Seg d st
( ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) & [(l . i0),(r . i0)] is Gap of G . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) ) ) implies cell (l,r) in cells (1,G) ) by Th43; ::_thesis: verum
end;

theorem Th45: :: CHAIN_1:45
for k, k9 being Element of NAT
for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k <= d & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
proof
let k, k9 be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k <= d & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
let d be non zero Element of NAT ; ::_thesis: for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k <= d & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
let l, r, l9, r9 be Element of REAL d; ::_thesis: for G being Grating of d st k <= d & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
let G be Grating of d; ::_thesis: ( k <= d & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) implies for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) )
assume that
A1: k <= d and
A2: k9 <= d and
A3: cell (l,r) in cells (k,G) and
A4: cell (l9,r9) in cells (k9,G) ; ::_thesis: ( not cell (l,r) c= cell (l9,r9) or for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) )
assume A5: cell (l,r) c= cell (l9,r9) ; ::_thesis: for i being Element of Seg d holds
( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
let i be Element of Seg d; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
percases ( for i being Element of Seg d holds
( ( l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( l9 . i = r9 . i & l9 . i in G . i ) ) or for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) by A2, A4, Th34;
supposeA6: for i being Element of Seg d holds
( ( l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( l9 . i = r9 . i & l9 . i in G . i ) ) ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
then A7: for i being Element of Seg d holds l9 . i <= r9 . i ;
then A8: l9 . i <= l . i by A5, Th28;
A9: l . i <= r . i by A5, A7, Th28;
A10: r . i <= r9 . i by A5, A7, Th28;
A11: l9 . i <= r . i by A8, A9, XXREAL_0:2;
A12: l . i <= r9 . i by A9, A10, XXREAL_0:2;
thus  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) ::_thesis: verum
proof
percases ( [(l9 . i),(r9 . i)] is Gap of G . i or l9 . i = r9 . i ) by A6;
supposeA13: [(l9 . i),(r9 . i)] is Gap of G . i ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
A14: now__::_thesis:_(_l9_._i_<>_l_._i_implies_not_l_._i_<>_r9_._i_)
assume that
A15: l9 . i <> l . i and
A16: l . i <> r9 . i ; ::_thesis: contradiction
A17: l9 . i < l . i by A8, A15, XXREAL_0:1;
A18: l . i < r9 . i by A12, A16, XXREAL_0:1;
 l . i in G . i by A1, A3, Th35;
hence  contradiction by A13, A17, A18, Th16; ::_thesis: verum
end;
now__::_thesis:_(_l9_._i_<>_r_._i_implies_not_r_._i_<>_r9_._i_)
assume that
A19: l9 . i <> r . i and
A20: r . i <> r9 . i ; ::_thesis: contradiction
A21: l9 . i < r . i by A11, A19, XXREAL_0:1;
A22: r . i < r9 . i by A10, A20, XXREAL_0:1;
 r . i in G . i by A1, A3, Th35;
hence  contradiction by A13, A21, A22, Th16; ::_thesis: verum
end;
hence  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) by A9, A14, XXREAL_0:1; ::_thesis: verum
end;
suppose l9 . i = r9 . i ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
hence  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) by A8, A10, A11, A12, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
supposeA23: for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
then A24: r9 . i < l9 . i ;
A25: [(l9 . i),(r9 . i)] is Gap of G . i by A23;
thus  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) ::_thesis: verum
proof
percases ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) or for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) ) by A1, A3, Th34;
supposeA26: for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
then A27: r . i <= r9 . i by A5, Th29;
A28: l9 . i <= l . i by A5, A26, Th29;
A29: now__::_thesis:_not_l9_._i_<>_l_._i
assume  l9 . i <> l . i ; ::_thesis: contradiction
then A30: l9 . i < l . i by A28, XXREAL_0:1;
 l . i in G . i by A1, A3, Th35;
hence  contradiction by A24, A25, A30, Th16; ::_thesis: verum
end;
now__::_thesis:_not_r_._i_<>_r9_._i
assume  r . i <> r9 . i ; ::_thesis: contradiction
then A31: r . i < r9 . i by A27, XXREAL_0:1;
 r . i in G . i by A1, A3, Th35;
hence  contradiction by A24, A25, A31, Th16; ::_thesis: verum
end;
hence  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) by A29; ::_thesis: verum
end;
supposeA32: for i being Element of Seg d holds
( ( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( l . i = r . i & l . i in G . i ) ) ; ::_thesis: ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) )
A33: l . i in G . i by A1, A3, Th35;
 r . i in G . i by A1, A3, Th35;
hence  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) or ( l . i <= r . i & r9 . i < l9 . i & r9 . i <= l . i & r . i <= l9 . i ) ) by A24, A25, A32, A33, Th16; ::_thesis: verum
end;
end;
end;
end;
end;
end;

theorem Th46: :: CHAIN_1:46
for k, k9 being Element of NAT
for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k < k9 & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
proof
let k, k9 be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k < k9 & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
let d be non zero Element of NAT ; ::_thesis: for l, r, l9, r9 being Element of REAL d
for G being Grating of d st k < k9 & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
let l, r, l9, r9 be Element of REAL d; ::_thesis: for G being Grating of d st k < k9 & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) holds
ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
let G be Grating of d; ::_thesis: ( k < k9 & k9 <= d & cell (l,r) in cells (k,G) & cell (l9,r9) in cells (k9,G) & cell (l,r) c= cell (l9,r9) implies ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) )
assume that
A1: k < k9 and
A2: k9 <= d and
A3: cell (l,r) in cells (k,G) and
A4: cell (l9,r9) in cells (k9,G) ; ::_thesis: ( not cell (l,r) c= cell (l9,r9) or ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) )
A5: k + 0 < d by A1, A2, XXREAL_0:2;
assume A6: cell (l,r) c= cell (l9,r9) ; ::_thesis: ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
consider X being Subset of (Seg d) such that
A7: card X = k and
A8: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) by A3, A5, Th33;
A9: d - k > 0 by A5, XREAL_1:20;
card ((Seg d) \ X) = (card (Seg d)) - (card X) by CARD_2:44
.= d - k by A7, FINSEQ_1:57 ;
then consider i0 being set  such that
A10: i0 in (Seg d) \ X by A9, CARD_1:27, XBOOLE_0:def_1;
reconsider i0 = i0 as Element of Seg d by A10, XBOOLE_0:def_5;
 not i0 in X by A10, XBOOLE_0:def_5;
then A11: l . i0 = r . i0 by A8;
percases ( ( l . i0 = l9 . i0 & r . i0 = r9 . i0 ) or ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) or ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) or r9 . i0 < l9 . i0 ) by A2, A3, A4, A5, A6, Th45;
suppose ( l . i0 = l9 . i0 & r . i0 = r9 . i0 ) ; ::_thesis: ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
hence  ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) by A11; ::_thesis: verum
end;
suppose ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) ; ::_thesis: ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
hence  ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) ; ::_thesis: verum
end;
suppose ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) ; ::_thesis: ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
hence  ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) ; ::_thesis: verum
end;
supposeA12: r9 . i0 < l9 . i0 ; ::_thesis: ex i being Element of Seg d st
( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
assume A13: for i being Element of Seg d holds
( ( l . i <> l9 . i or r . i <> l9 . i ) & ( l . i <> r9 . i or r . i <> r9 . i ) ) ; ::_thesis: contradiction
defpred S1[ Element of Seg d, Real] means ( l . $1 <= $2 & $2 <= r . $1 & r9 . $1 < $2 & $2 < l9 . $1 );
A14: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
A15: l . i in G . i by A3, A5, Th35;
A16: r . i in G . i by A3, A5, Th35;
A17: r9 . i < l9 . i by A2, A4, A12, Th34;
A18: [(l9 . i),(r9 . i)] is Gap of G . i by A2, A4, A12, Th34;
percases ( ( r9 . i < l . i & l . i < l9 . i ) or l . i <= r9 . i or l9 . i <= l . i ) ;
supposeA19: ( r9 . i < l . i & l . i < l9 . i ) ; ::_thesis: ex xi being Real st S1[i,xi]
take  l . i ; ::_thesis: S1[i,l . i]
thus  S1[i,l . i] by A8, A19; ::_thesis: verum
end;
supposeA20: l . i <= r9 . i ; ::_thesis: ex xi being Real st S1[i,xi]
A21: l . i >= r9 . i by A15, A17, A18, Th16;
then A22: l . i = r9 . i by A20, XXREAL_0:1;
then  r . i <> r9 . i by A13;
then  l . i < r . i by A8, A22;
then consider xi being Real such that
A23: l . i < xi and
A24: xi < r . i by Th1;
take  xi ; ::_thesis: S1[i,xi]
 r . i <= l9 . i by A16, A17, A18, Th16;
hence  S1[i,xi] by A21, A23, A24, XXREAL_0:2; ::_thesis: verum
end;
supposeA25: l9 . i <= l . i ; ::_thesis: ex xi being Real st S1[i,xi]
 l9 . i >= l . i by A15, A17, A18, Th16;
then A26: l9 . i = l . i by A25, XXREAL_0:1;
 l9 . i >= r . i by A16, A17, A18, Th16;
then  l9 . i = r . i by A8, A26;
hence  ex xi being Real st S1[i,xi] by A13, A26; ::_thesis: verum
end;
end;
end;
consider x being Function of (Seg d),REAL such that
A27: for i being Element of Seg d holds S1[i,x . i] from FUNCT_2:sch_3(A14);
reconsider x = x as Element of REAL d by Def3;
A28: x in cell (l,r) by A27;
 for i being Element of Seg d st r9 . i < l9 . i holds
( r9 . i < x . i & x . i < l9 . i ) by A27;
hence  contradiction by A6, A12, A28, Th25; ::_thesis: verum
end;
end;
end;

theorem Th47: :: CHAIN_1:47
for d being non zero Element of NAT
for l, r, l9, r9 being Element of REAL d
for G being Grating of d
for X, X9 being Subset of (Seg d) st cell (l,r) c= cell (l9,r9) & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) & ( for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ) holds
( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r, l9, r9 being Element of REAL d
for G being Grating of d
for X, X9 being Subset of (Seg d) st cell (l,r) c= cell (l9,r9) & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) & ( for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ) holds
( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
let l, r, l9, r9 be Element of REAL d; ::_thesis: for G being Grating of d
for X, X9 being Subset of (Seg d) st cell (l,r) c= cell (l9,r9) & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) & ( for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ) holds
( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
let G be Grating of d; ::_thesis: for X, X9 being Subset of (Seg d) st cell (l,r) c= cell (l9,r9) & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) & ( for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ) holds
( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
let X, X9 be Subset of (Seg d); ::_thesis: ( cell (l,r) c= cell (l9,r9) & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) & ( for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ) implies ( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) ) )
assume A1: cell (l,r) c= cell (l9,r9) ; ::_thesis: ( ex i being Element of Seg d st
( not ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) & not ( not i in X & l . i = r . i & l . i in G . i ) ) or ex i being Element of Seg d st
( not ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) & not ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) or ( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) ) )
assume A2: for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ; ::_thesis: ( ex i being Element of Seg d st
( not ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) & not ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) or ( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) ) )
assume A3: for i being Element of Seg d holds
( ( i in X9 & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X9 & l9 . i = r9 . i & l9 . i in G . i ) ) ; ::_thesis: ( X c= X9 & ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
A4: l in cell (l,r) by Th26;
A5: r in cell (l,r) by Th26;
A6: for i being Element of Seg d holds l9 . i <= r9 . i by A3;
thus  X c= X9 ::_thesis: ( ( for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ) & ( for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ) )
proof
let i be set ; :: according to TARSKI:def_3 ::_thesis: ( not i in X or i in X9 )
assume that
A7: i in X and
A8: not i in X9 ; ::_thesis: contradiction
reconsider i = i as Element of Seg d by A7;
A9: l . i < r . i by A2, A7;
A10: l9 . i = r9 . i by A3, A8;
A11: l9 . i <= l . i by A1, A4, A6, Th24;
 r . i <= r9 . i by A1, A5, A6, Th24;
hence  contradiction by A9, A10, A11, XXREAL_0:2; ::_thesis: verum
end;
set k = card X;
set k9 = card X9;
A12: card (Seg d) = d by FINSEQ_1:57;
then A13: card X <= d by NAT_1:43;
A14: card X9 <= d by A12, NAT_1:43;
A15: cell (l,r) in cells ((card X),G) by A2, A13, Th33;
A16: cell (l9,r9) in cells ((card X9),G) by A3, A14, Th33;
thus  for i being Element of Seg d st ( i in X or not i in X9 ) holds
( l . i = l9 . i & r . i = r9 . i ) ::_thesis: for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i )
proof
let i be Element of Seg d; ::_thesis: ( ( i in X or not i in X9 ) implies ( l . i = l9 . i & r . i = r9 . i ) )
assume A17: ( i in X or not i in X9 ) ; ::_thesis: ( l . i = l9 . i & r . i = r9 . i )
 l9 . i <= r9 . i by A3;
then  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) by A1, A13, A14, A15, A16, Th45;
hence  ( l . i = l9 . i & r . i = r9 . i ) by A2, A3, A17; ::_thesis: verum
end;
thus  for i being Element of Seg d st not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) holds
( l . i = r9 . i & r . i = r9 . i ) ::_thesis: verum
proof
let i be Element of Seg d; ::_thesis: ( not i in X & i in X9 & not ( l . i = l9 . i & r . i = l9 . i ) implies ( l . i = r9 . i & r . i = r9 . i ) )
assume that
A18: not i in X and
A19: i in X9 ; ::_thesis: ( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) )
A20: l . i = r . i by A2, A18;
 l9 . i < r9 . i by A3, A19;
hence  ( ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) by A1, A13, A14, A15, A16, A20, Th45; ::_thesis: verum
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
mode Cell of k,G is Element of cells (k,G);
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
mode Chain of k,G is Subset of (cells (k,G));
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
func 0_ (k,G) ->  Chain of k,G equals :: CHAIN_1:def 8
 {} ;
coherence
 {} is Chain of k,G by SUBSET_1:1;
end;

:: deftheorem  defines 0_ CHAIN_1:def_8_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT holds 0_ (k,G) = {} ;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
func Omega G ->  Chain of d,G equals :: CHAIN_1:def 9
 cells (d,G);
coherence
 cells (d,G) is Chain of d,G
proof
 cells (d,G) c= cells (d,G) ;
hence  cells (d,G) is Chain of d,G ; ::_thesis: verum
end;
end;

:: deftheorem  defines Omega CHAIN_1:def_9_:_
 for d being non zero Element of NAT
for G being Grating of d holds Omega G = cells (d,G);

notation
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C1, C2 be Chain of k,G;
synonym C1 + C2 for d \+\ G;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C1, C2 be Chain of k,G;
:: original: +
redefine funcC1 + C2 ->  Chain of k,G;
coherence
 + is Chain of k,G
proof
 C1 \+\ C2 c= cells (k,G) ;
hence  + is Chain of k,G ; ::_thesis: verum
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
func infinite-cell G ->  Cell of d,G means :Def10: :: CHAIN_1:def 10
 ex l, r being Element of REAL d st
( it = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) );
existence
 ex b1 being Cell of d,G ex l, r being Element of REAL d st
( b1 = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) )
proof
defpred S1[ Element of Seg d, Real] means ( $2 in G . $1 & ( for xi being Real st xi in G . $1 holds
xi <= $2 ) );
A1: for i being Element of Seg d ex li being Real st S1[i,li] by Th11;
consider l being Function of (Seg d),REAL such that
A2: for i being Element of Seg d holds S1[i,l . i] from FUNCT_2:sch_3(A1);
reconsider l = l as Element of REAL d by Def3;
defpred S2[ Element of Seg d, Real] means ( $2 in G . $1 & ( for xi being Real st xi in G . $1 holds
xi >= $2 ) );
A3: for i being Element of Seg d ex ri being Real st S2[i,ri] by Th12;
consider r being Function of (Seg d),REAL such that
A4: for i being Element of Seg d holds S2[i,r . i] from FUNCT_2:sch_3(A3);
reconsider r = r as Element of REAL d by Def3;
A5: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_r_._i_<_l_._i
let i be Element of Seg d; ::_thesis: r . i < l . i
 r . i in G . i by A4;
then A6: r . i <= l . i by A2;
now__::_thesis:_not_l_._i_=_r_._i
assume A7: l . i = r . i ; ::_thesis: contradiction
consider x1, x2 being set  such that
A8: x1 in G . i and
A9: x2 in G . i and
A10: x1 <> x2 by ZFMISC_1:def_10;
reconsider x1 = x1, x2 = x2 as Real by A8, A9;
A11: r . i <= x1 by A4, A8;
A12: x1 <= l . i by A2, A8;
A13: r . i <= x2 by A4, A9;
A14: x2 <= l . i by A2, A9;
 x1 = l . i by A7, A11, A12, XXREAL_0:1;
hence  contradiction by A7, A10, A13, A14, XXREAL_0:1; ::_thesis: verum
end;
hence  r . i < l . i by A6, XXREAL_0:1; ::_thesis: verum
end;
A15: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_r_._i_<_l_._i_&_[(l_._i),(r_._i)]_is_Gap_of_G_._i_)
let i be Element of Seg d; ::_thesis: ( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
A16: l . i in G . i by A2;
A17: r . i in G . i by A4;
A18: r . i < l . i by A5;
 for xi being Real st xi in G . i holds
( not l . i < xi & not xi < r . i ) by A2, A4;
hence  ( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A16, A17, A18, Th16; ::_thesis: verum
end;
then reconsider A = cell (l,r) as Cell of d,G by Th33;
take  A ; ::_thesis: ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) )
take  l ; ::_thesis: ex r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) )
take  r ; ::_thesis: ( A = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) )
thus  ( A = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) by A15; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Cell of d,G st ex l, r being Element of REAL d st
( b1 = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) & ex l, r being Element of REAL d st
( b2 = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) holds
b1 = b2
proof
let A, A9 be Cell of d,G; ::_thesis: ( ex l, r being Element of REAL d st
( A = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) & ex l, r being Element of REAL d st
( A9 = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) implies A = A9 )
given l, r being Element of REAL d such that A19: A = cell (l,r) and
A20: for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ; ::_thesis: ( for l, r being Element of REAL d holds
( not A9 = cell (l,r) or ex i being Element of Seg d st
( r . i < l . i implies not [(l . i),(r . i)] is Gap of G . i ) ) or A = A9 )
given l9, r9 being Element of REAL d such that A21: A9 = cell (l9,r9) and
A22: for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ; ::_thesis: A = A9
reconsider l = l, r = r, l9 = l9, r9 = r9 as Function of (Seg d),REAL by Def3;
A23: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l_._i_=_l9_._i_&_r_._i_=_r9_._i_)
let i be Element of Seg d; ::_thesis: ( l . i = l9 . i & r . i = r9 . i )
A24: r . i < l . i by A20;
A25: [(l . i),(r . i)] is Gap of G . i by A20;
A26: r9 . i < l9 . i by A22;
 [(l9 . i),(r9 . i)] is Gap of G . i by A22;
hence  ( l . i = l9 . i & r . i = r9 . i ) by A24, A25, A26, Th22; ::_thesis: verum
end;
then  l = l9 by FUNCT_2:63;
hence  A = A9 by A19, A21, A23, FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def10 defines infinite-cell CHAIN_1:def_10_:_
 for d being non zero Element of NAT
for G being Grating of d
for b3 being Cell of d,G holds
( b3 = infinite-cell G iff ex l, r being Element of REAL d st
( b3 = cell (l,r) & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) );

theorem Th48: :: CHAIN_1:48
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d st cell (l,r) is Cell of d,G holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d st cell (l,r) is Cell of d,G holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d st cell (l,r) is Cell of d,G holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
let G be Grating of d; ::_thesis: ( cell (l,r) is Cell of d,G implies ( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i ) )
assume A1: cell (l,r) is Cell of d,G ; ::_thesis: ( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds r . i < l . i )
then reconsider A = cell (l,r) as Cell of d,G ;
hereby  ::_thesis: ( ( for i being Element of Seg d holds r . i < l . i ) implies cell (l,r) = infinite-cell G )
assume  cell (l,r) = infinite-cell G ; ::_thesis: for i being Element of Seg d holds r . i < l . i
then consider l9, r9 being Element of REAL d such that
A2: cell (l,r) = cell (l9,r9) and
A3: for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) by Def10;
A4: l = l9 by A2, A3, Th31;
 r = r9 by A2, A3, Th31;
hence  for i being Element of Seg d holds r . i < l . i by A3, A4; ::_thesis: verum
end;
set i0 = the Element of Seg d;
assume  for i being Element of Seg d holds r . i < l . i ; ::_thesis: cell (l,r) = infinite-cell G
then A5: r . the Element of Seg d < l . the Element of Seg d ;
A6: A = cell (l,r) ;
 for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A1, A5, Th34;
hence  cell (l,r) = infinite-cell G by A6, Def10; ::_thesis: verum
end;

theorem Th49: :: CHAIN_1:49
for d being non zero Element of NAT
for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
proof
let d be non zero Element of NAT ; ::_thesis: for l, r being Element of REAL d
for G being Grating of d holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
let l, r be Element of REAL d; ::_thesis: for G being Grating of d holds
( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
let G be Grating of d; ::_thesis: ( cell (l,r) = infinite-cell G iff for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) )
hereby  ::_thesis: ( ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) implies cell (l,r) = infinite-cell G )
assume  cell (l,r) = infinite-cell G ; ::_thesis: for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i )
then consider l9, r9 being Element of REAL d such that
A1: cell (l,r) = cell (l9,r9) and
A2: for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) by Def10;
A3: l = l9 by A1, A2, Th31;
 r = r9 by A1, A2, Th31;
hence  for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) by A2, A3; ::_thesis: verum
end;
assume A4: for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ; ::_thesis: cell (l,r) = infinite-cell G
then  cell (l,r) is Cell of d,G by Th33;
hence  cell (l,r) = infinite-cell G by A4, Def10; ::_thesis: verum
end;

scheme :: CHAIN_1:sch 4
ChainInd{ F1() ->  non zero Element of NAT , F2() ->  Grating of F1(), F3() ->  Element of NAT , F4() ->  Chain of F3(),F2(), P1[ set ] } :
 P1[F4()]
provided
A1: P1[ 0_ (F3(),F2())] and
A2: for A being Cell of F3(),F2() st A in F4() holds
P1[{A}] and
A3: for C1, C2 being Chain of F3(),F2() st C1 c= F4() & C2 c= F4() & P1[C1] & P1[C2] holds
P1[C1 + C2]
proof
A4: F4() is finite ;
A5: P1[ {} ] by A1;
A6: for x, B being set st x in F4() & B c= F4() & P1[B] holds
P1[B \/ {x}]
proof
let A, C1 be set ; ::_thesis: ( A in F4() & C1 c= F4() & P1[C1] implies P1[C1 \/ {A}] )
assume that
A7: A in F4() and
A8: C1 c= F4() and
A9: P1[C1] ; ::_thesis: P1[C1 \/ {A}]
reconsider A9 = A as Cell of F3(),F2() by A7;
reconsider C19 = C1 as Chain of F3(),F2() by A8, XBOOLE_1:1;
percases ( A in C1 or not A in C1 ) ;
suppose A in C1 ; ::_thesis: P1[C1 \/ {A}]
then  {A} c= C1 by ZFMISC_1:31;
hence  P1[C1 \/ {A}] by A9, XBOOLE_1:12; ::_thesis: verum
end;
supposeA10: not A in C1 ; ::_thesis: P1[C1 \/ {A}]
now__::_thesis:_for_A9_being_set_holds_not_A9_in_C1_/\_{A}
let A9 be set ; ::_thesis: not A9 in C1 /\ {A}
assume A11: A9 in C1 /\ {A} ; ::_thesis: contradiction
then A12: A9 in C1 by XBOOLE_0:def_4;
 A9 in {A} by A11, XBOOLE_0:def_4;
hence  contradiction by A10, A12, TARSKI:def_1; ::_thesis: verum
end;
then  C1 /\ {A} = {} by XBOOLE_0:def_1;
then A13: C19 + {A9} = (C1 \/ {A}) \ {} by XBOOLE_1:101
.= C1 \/ {A} ;
A14: P1[{A9}] by A2, A7;
 {A} c= F4() by A7, ZFMISC_1:31;
hence  P1[C1 \/ {A}] by A3, A8, A9, A13, A14; ::_thesis: verum
end;
end;
end;
thus  P1[F4()] from FINSET_1:sch_2(A4, A5, A6); ::_thesis: verum
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let A be Cell of k,G;
func star A ->  Chain of (k + 1),G equals :: CHAIN_1:def 11
 {  B where B is Cell of (k + 1),G : A c= B  }  ;
coherence
  {  B where B is Cell of (k + 1),G : A c= B  }  is Chain of (k + 1),G
proof
defpred S1[ set ] means A c= $1;
  {  B where B is Cell of (k + 1),G : S1[B]  }  c= cells ((k + 1),G) from FRAENKEL:sch_10();
hence   {  B where B is Cell of (k + 1),G : A c= B  }  is Chain of (k + 1),G ; ::_thesis: verum
end;
end;

:: deftheorem  defines star CHAIN_1:def_11_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for A being Cell of k,G holds star A =  {  B where B is Cell of (k + 1),G : A c= B  }  ;

theorem Th50: :: CHAIN_1:50
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( B in star A iff A c= B )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( B in star A iff A c= B )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( B in star A iff A c= B )
let G be Grating of d; ::_thesis: for A being Cell of k,G
for B being Cell of (k + 1),G holds
( B in star A iff A c= B )
let A be Cell of k,G; ::_thesis: for B being Cell of (k + 1),G holds
( B in star A iff A c= B )
let B be Cell of (k + 1),G; ::_thesis: ( B in star A iff A c= B )
defpred S1[ set ] means A c= $1;
A1: star A =  {  B9 where B9 is Cell of (k + 1),G : S1[B9]  }  ;
thus  ( B in star A iff S1[B] ) from LMOD_7:sch_7(A1); ::_thesis: verum
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C be Chain of (k + 1),G;
func del C ->  Chain of k,G equals :: CHAIN_1:def 12
 {  A where A is Cell of k,G : ( k + 1 <= d & card ((star A) /\ C) is odd )  }  ;
coherence
  {  A where A is Cell of k,G : ( k + 1 <= d & card ((star A) /\ C) is odd )  }  is Chain of k,G
proof
defpred S1[ Cell of k,G] means ( k + 1 <= d & card ((star $1) /\ C) is odd );
  {  A where A is Cell of k,G : S1[A]  }  c= cells (k,G) from FRAENKEL:sch_10();
hence   {  A where A is Cell of k,G : ( k + 1 <= d & card ((star A) /\ C) is odd )  }  is Chain of k,G ; ::_thesis: verum
end;
end;

:: deftheorem  defines del CHAIN_1:def_12_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for C being Chain of (k + 1),G holds del C =  {  A where A is Cell of k,G : ( k + 1 <= d & card ((star A) /\ C) is odd )  }  ;

notation
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C be Chain of (k + 1),G;
synonym  . C for  del C;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C be Chain of (k + 1),G;
let C9 be Chain of k,G;
predC9 bounds C means :: CHAIN_1:def 13
C9 = del C;
end;

:: deftheorem  defines bounds CHAIN_1:def_13_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for C being Chain of (k + 1),G
for C9 being Chain of k,G holds
( C9 bounds C iff C9 = del C );

theorem Th51: :: CHAIN_1:51
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for A being Cell of k,G
for C being Chain of (k + 1),G holds
( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d
for A being Cell of k,G
for C being Chain of (k + 1),G holds
( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for A being Cell of k,G
for C being Chain of (k + 1),G holds
( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
let G be Grating of d; ::_thesis: for A being Cell of k,G
for C being Chain of (k + 1),G holds
( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
let A be Cell of k,G; ::_thesis: for C being Chain of (k + 1),G holds
( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
let C be Chain of (k + 1),G; ::_thesis: ( A in del C iff ( k + 1 <= d & card ((star A) /\ C) is odd ) )
defpred S1[ Cell of k,G] means ( k + 1 <= d & card ((star $1) /\ C) is odd );
A1: del C =  {  A9 where A9 is Cell of k,G : S1[A9]  }  ;
thus  ( A in del C iff S1[A] ) from LMOD_7:sch_7(A1); ::_thesis: verum
end;

theorem Th52: :: CHAIN_1:52
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st k + 1 > d holds
for C being Chain of (k + 1),G holds del C = 0_ (k,G)
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st k + 1 > d holds
for C being Chain of (k + 1),G holds del C = 0_ (k,G)
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st k + 1 > d holds
for C being Chain of (k + 1),G holds del C = 0_ (k,G)
let G be Grating of d; ::_thesis: ( k + 1 > d implies for C being Chain of (k + 1),G holds del C = 0_ (k,G) )
assume A1: k + 1 > d ; ::_thesis: for C being Chain of (k + 1),G holds del C = 0_ (k,G)
let C be Chain of (k + 1),G; ::_thesis: del C = 0_ (k,G)
 for A being set holds not A in del C by A1, Th51;
hence  del C = 0_ (k,G) by XBOOLE_0:def_1; ::_thesis: verum
end;

theorem Th53: :: CHAIN_1:53
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st k + 1 <= d holds
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st k + 1 <= d holds
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st k + 1 <= d holds
for A being Cell of k,G
for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B )
let G be Grating of d; ::_thesis: ( k + 1 <= d implies for A being Cell of k,G
for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B ) )
assume A1: k + 1 <= d ; ::_thesis: for A being Cell of k,G
for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B )
let A be Cell of k,G; ::_thesis: for B being Cell of (k + 1),G holds
( A in del {B} iff A c= B )
let B be Cell of (k + 1),G; ::_thesis: ( A in del {B} iff A c= B )
set X = (star A) /\ {B};
 ( card ((star A) /\ {B}) is odd iff B in star A )
proof
percases ( B in star A or not B in star A ) ;
supposeA2: B in star A ; ::_thesis: ( card ((star A) /\ {B}) is odd iff B in star A )
now__::_thesis:_for_B9_being_set_holds_
(_B9_in_(star_A)_/\_{B}_iff_B9_=_B_)
let B9 be set ; ::_thesis: ( B9 in (star A) /\ {B} iff B9 = B )
 ( B9 in {B} iff B9 = B ) by TARSKI:def_1;
hence  ( B9 in (star A) /\ {B} iff B9 = B ) by A2, XBOOLE_0:def_4; ::_thesis: verum
end;
then  (star A) /\ {B} = {B} by TARSKI:def_1;
then  card ((star A) /\ {B}) = (2 * 0) + 1 by CARD_1:30;
hence  ( card ((star A) /\ {B}) is odd iff B in star A ) by A2; ::_thesis: verum
end;
supposeA3: not B in star A ; ::_thesis: ( card ((star A) /\ {B}) is odd iff B in star A )
now__::_thesis:_for_B9_being_set_holds_not_B9_in_(star_A)_/\_{B}
let B9 be set ; ::_thesis: not B9 in (star A) /\ {B}
 ( B9 = B or not B9 in {B} ) by TARSKI:def_1;
hence  not B9 in (star A) /\ {B} by A3, XBOOLE_0:def_4; ::_thesis: verum
end;
then  card ((star A) /\ {B}) = 2 * 0 by CARD_1:27, XBOOLE_0:def_1;
hence  ( card ((star A) /\ {B}) is odd iff B in star A ) by A3; ::_thesis: verum
end;
end;
end;
hence  ( A in del {B} iff A c= B ) by A1, Th50, Th51; ::_thesis: verum
end;

theorem Th54: :: CHAIN_1:54
for d9 being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st d = d9 + 1 holds
for A being Cell of d9,G holds card (star A) = 2
proof
let d9 be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st d = d9 + 1 holds
for A being Cell of d9,G holds card (star A) = 2
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st d = d9 + 1 holds
for A being Cell of d9,G holds card (star A) = 2
let G be Grating of d; ::_thesis: ( d = d9 + 1 implies for A being Cell of d9,G holds card (star A) = 2 )
assume A1: d = d9 + 1 ; ::_thesis: for A being Cell of d9,G holds card (star A) = 2
then A2: d9 < d by NAT_1:13;
let A be Cell of d9,G; ::_thesis: card (star A) = 2
consider l, r being Element of REAL d, i0 being Element of Seg d such that
A3: A = cell (l,r) and
A4: l . i0 = r . i0 and
A5: l . i0 in G . i0 and
A6: for i being Element of Seg d st i <> i0 holds
( l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) by A1, Th41;
A7: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l_._i_<=_r_._i
let i be Element of Seg d; ::_thesis: l . i <= r . i
 ( i = i0 or i <> i0 ) ;
hence  l . i <= r . i by A4, A6; ::_thesis: verum
end;
 ex B1, B2 being set st
( B1 in star A & B2 in star A & B1 <> B2 & ( for B being set holds
( not B in star A or B = B1 or B = B2 ) ) )
proof
 ex l1, r1 being Element of REAL d st
( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
proof
consider l1i0 being Real such that
A8: [l1i0,(l . i0)] is Gap of G . i0 by A5, Th19;
percases ( l1i0 < l . i0 or l . i0 < l1i0 ) by A8, Th16;
supposeA9: l1i0 < l . i0 ; ::_thesis: ex l1, r1 being Element of REAL d st
( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
defpred S1[ Element of Seg d, Real] means ( ( $1 = i0 implies $2 = l1i0 ) & ( $1 <> i0 implies $2 = l . $1 ) );
A10: for i being Element of Seg d ex li being Real st S1[i,li]
proof
let i be Element of Seg d; ::_thesis: ex li being Real st S1[i,li]
 ( i = i0 or i <> i0 ) ;
hence  ex li being Real st S1[i,li] ; ::_thesis: verum
end;
consider l1 being Function of (Seg d),REAL such that
A11: for i being Element of Seg d holds S1[i,l1 . i] from FUNCT_2:sch_3(A10);
reconsider l1 = l1 as Element of REAL d by Def3;
take  l1 ; ::_thesis: ex r1 being Element of REAL d st
( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
take  r ; ::_thesis: ( [(l1 . i0),(r . i0)] is Gap of G . i0 & r . i0 = l . i0 & ( ( l1 . i0 < r . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r . i = r . i ) ) ) or for i being Element of Seg d holds
( r . i < l1 . i & [(l1 . i),(r . i)] is Gap of G . i ) ) )
thus  ( [(l1 . i0),(r . i0)] is Gap of G . i0 & r . i0 = l . i0 & ( ( l1 . i0 < r . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r . i = r . i ) ) ) or for i being Element of Seg d holds
( r . i < l1 . i & [(l1 . i),(r . i)] is Gap of G . i ) ) ) by A4, A8, A9, A11; ::_thesis: verum
end;
supposeA12: l . i0 < l1i0 ; ::_thesis: ex l1, r1 being Element of REAL d st
( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
consider l1, r1 being Element of REAL d such that
 cell (l1,r1) = infinite-cell G and
A13: for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) by Def10;
take  l1 ; ::_thesis: ex r1 being Element of REAL d st
( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
take  r1 ; ::_thesis: ( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) )
A14: r1 . i0 < l1 . i0 by A13;
 [(l1 . i0),(r1 . i0)] is Gap of G . i0 by A13;
hence  ( [(l1 . i0),(r1 . i0)] is Gap of G . i0 & r1 . i0 = l . i0 & ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) ) by A8, A12, A13, A14, Th22; ::_thesis: verum
end;
end;
end;
then consider l1, r1 being Element of REAL d such that
A15: [(l1 . i0),(r1 . i0)] is Gap of G . i0 and
A16: r1 . i0 = l . i0 and
A17: ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) ;
A18: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_[(l1_._i),(r1_._i)]_is_Gap_of_G_._i
let i be Element of Seg d; ::_thesis: [(l1 . i),(r1 . i)] is Gap of G . i
A19: ( i <> i0 & l1 . i = l . i & r1 . i = r . i implies [(l1 . i),(r1 . i)] is Gap of G . i ) by A6;
 ( i = i0 or i <> i0 ) ;
hence  [(l1 . i),(r1 . i)] is Gap of G . i by A15, A17, A19; ::_thesis: verum
end;
A20: ( for i being Element of Seg d holds l1 . i < r1 . i or for i being Element of Seg d holds r1 . i < l1 . i )
proof
percases ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) by A17;
supposeA21: ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) ; ::_thesis: ( for i being Element of Seg d holds l1 . i < r1 . i or for i being Element of Seg d holds r1 . i < l1 . i )
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l1_._i_<_r1_._i
let i be Element of Seg d; ::_thesis: l1 . i < r1 . i
A22: ( i <> i0 & l1 . i = l . i & r1 . i = r . i implies l1 . i < r1 . i ) by A6;
 ( i = i0 or i <> i0 ) ;
hence  l1 . i < r1 . i by A21, A22; ::_thesis: verum
end;
hence  ( for i being Element of Seg d holds l1 . i < r1 . i or for i being Element of Seg d holds r1 . i < l1 . i ) ; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ; ::_thesis: ( for i being Element of Seg d holds l1 . i < r1 . i or for i being Element of Seg d holds r1 . i < l1 . i )
hence  ( for i being Element of Seg d holds l1 . i < r1 . i or for i being Element of Seg d holds r1 . i < l1 . i ) ; ::_thesis: verum
end;
end;
end;
then reconsider B1 = cell (l1,r1) as Cell of d,G by A18, Th40;
 ex l2, r2 being Element of REAL d st
( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) )
proof
consider r2i0 being Real such that
A23: [(l . i0),r2i0] is Gap of G . i0 by A5, Th18;
percases ( l . i0 < r2i0 or r2i0 < l . i0 ) by A23, Th16;
supposeA24: l . i0 < r2i0 ; ::_thesis: ex l2, r2 being Element of REAL d st
( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) )
defpred S1[ Element of Seg d, Real] means ( ( $1 = i0 implies $2 = r2i0 ) & ( $1 <> i0 implies $2 = r . $1 ) );
A25: for i being Element of Seg d ex ri being Real st S1[i,ri]
proof
let i be Element of Seg d; ::_thesis: ex ri being Real st S1[i,ri]
 ( i = i0 or i <> i0 ) ;
hence  ex ri being Real st S1[i,ri] ; ::_thesis: verum
end;
consider r2 being Function of (Seg d),REAL such that
A26: for i being Element of Seg d holds S1[i,r2 . i] from FUNCT_2:sch_3(A25);
reconsider r2 = r2 as Element of REAL d by Def3;
take  l ; ::_thesis: ex r2 being Element of REAL d st
( [(l . i0),(r2 . i0)] is Gap of G . i0 & l . i0 = l . i0 & ( ( l . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l . i & [(l . i),(r2 . i)] is Gap of G . i ) ) )
take  r2 ; ::_thesis: ( [(l . i0),(r2 . i0)] is Gap of G . i0 & l . i0 = l . i0 & ( ( l . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l . i & [(l . i),(r2 . i)] is Gap of G . i ) ) )
thus  ( [(l . i0),(r2 . i0)] is Gap of G . i0 & l . i0 = l . i0 & ( ( l . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l . i & [(l . i),(r2 . i)] is Gap of G . i ) ) ) by A23, A24, A26; ::_thesis: verum
end;
supposeA27: r2i0 < l . i0 ; ::_thesis: ex l2, r2 being Element of REAL d st
( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) )
consider l2, r2 being Element of REAL d such that
 cell (l2,r2) = infinite-cell G and
A28: for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) by Def10;
take  l2 ; ::_thesis: ex r2 being Element of REAL d st
( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) )
take  r2 ; ::_thesis: ( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) )
A29: r2 . i0 < l2 . i0 by A28;
 [(l2 . i0),(r2 . i0)] is Gap of G . i0 by A28;
hence  ( [(l2 . i0),(r2 . i0)] is Gap of G . i0 & l2 . i0 = l . i0 & ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) ) by A23, A27, A28, A29, Th22; ::_thesis: verum
end;
end;
end;
then consider l2, r2 being Element of REAL d such that
A30: [(l2 . i0),(r2 . i0)] is Gap of G . i0 and
A31: l2 . i0 = l . i0 and
A32: ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) ;
A33: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_[(l2_._i),(r2_._i)]_is_Gap_of_G_._i
let i be Element of Seg d; ::_thesis: [(l2 . i),(r2 . i)] is Gap of G . i
A34: ( i <> i0 & l2 . i = l . i & r2 . i = r . i implies [(l2 . i),(r2 . i)] is Gap of G . i ) by A6;
 ( i = i0 or i <> i0 ) ;
hence  [(l2 . i),(r2 . i)] is Gap of G . i by A30, A32, A34; ::_thesis: verum
end;
 ( for i being Element of Seg d holds l2 . i < r2 . i or for i being Element of Seg d holds r2 . i < l2 . i )
proof
percases ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) by A32;
supposeA35: ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) ; ::_thesis: ( for i being Element of Seg d holds l2 . i < r2 . i or for i being Element of Seg d holds r2 . i < l2 . i )
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l2_._i_<_r2_._i
let i be Element of Seg d; ::_thesis: l2 . i < r2 . i
A36: ( i <> i0 & l2 . i = l . i & r2 . i = r . i implies l2 . i < r2 . i ) by A6;
 ( i = i0 or i <> i0 ) ;
hence  l2 . i < r2 . i by A35, A36; ::_thesis: verum
end;
hence  ( for i being Element of Seg d holds l2 . i < r2 . i or for i being Element of Seg d holds r2 . i < l2 . i ) ; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ; ::_thesis: ( for i being Element of Seg d holds l2 . i < r2 . i or for i being Element of Seg d holds r2 . i < l2 . i )
hence  ( for i being Element of Seg d holds l2 . i < r2 . i or for i being Element of Seg d holds r2 . i < l2 . i ) ; ::_thesis: verum
end;
end;
end;
then reconsider B2 = cell (l2,r2) as Cell of d,G by A33, Th40;
take  B1 ; ::_thesis: ex B2 being set st
( B1 in star A & B2 in star A & B1 <> B2 & ( for B being set holds
( not B in star A or B = B1 or B = B2 ) ) )
take  B2 ; ::_thesis: ( B1 in star A & B2 in star A & B1 <> B2 & ( for B being set holds
( not B in star A or B = B1 or B = B2 ) ) )
 A c= B1
proof
percases ( ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) or for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ) by A17;
supposeA37: ( l1 . i0 < r1 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l1 . i = l . i & r1 . i = r . i ) ) ) ; ::_thesis: A c= B1
A38: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l1_._i_<=_r1_._i
let i be Element of Seg d; ::_thesis: l1 . i <= r1 . i
 ( i = i0 or ( i <> i0 & l1 . i = l . i & r1 . i = r . i ) ) by A37;
hence  l1 . i <= r1 . i by A6, A37; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l1_._i_<=_l_._i_&_l_._i_<=_r_._i_&_r_._i_<=_r1_._i_)
let i be Element of Seg d; ::_thesis: ( l1 . i <= l . i & l . i <= r . i & r . i <= r1 . i )
 ( i = i0 or ( i <> i0 & l1 . i = l . i & r1 . i = r . i ) ) by A37;
hence  ( l1 . i <= l . i & l . i <= r . i & r . i <= r1 . i ) by A4, A16, A38; ::_thesis: verum
end;
hence  A c= B1 by A3, A38, Th28; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r1 . i < l1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) ; ::_thesis: A c= B1
hence  A c= B1 by A3, A4, A7, A16, Th30; ::_thesis: verum
end;
end;
end;
hence  B1 in star A by A1; ::_thesis: ( B2 in star A & B1 <> B2 & ( for B being set holds
( not B in star A or B = B1 or B = B2 ) ) )
 A c= B2
proof
percases ( ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) or for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ) by A32;
supposeA39: ( l2 . i0 < r2 . i0 & ( for i being Element of Seg d st i <> i0 holds
( l2 . i = l . i & r2 . i = r . i ) ) ) ; ::_thesis: A c= B2
A40: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l2_._i_<=_r2_._i
let i be Element of Seg d; ::_thesis: l2 . i <= r2 . i
 ( i = i0 or ( i <> i0 & l2 . i = l . i & r2 . i = r . i ) ) by A39;
hence  l2 . i <= r2 . i by A6, A39; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l2_._i_<=_l_._i_&_l_._i_<=_r_._i_&_r_._i_<=_r2_._i_)
let i be Element of Seg d; ::_thesis: ( l2 . i <= l . i & l . i <= r . i & r . i <= r2 . i )
 ( i = i0 or ( i <> i0 & l2 . i = l . i & r2 . i = r . i ) ) by A39;
hence  ( l2 . i <= l . i & l . i <= r . i & r . i <= r2 . i ) by A4, A31, A40; ::_thesis: verum
end;
hence  A c= B2 by A3, A40, Th28; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r2 . i < l2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) ; ::_thesis: A c= B2
hence  A c= B2 by A3, A7, A31, Th30; ::_thesis: verum
end;
end;
end;
hence  B2 in star A by A1; ::_thesis: ( B1 <> B2 & ( for B being set holds
( not B in star A or B = B1 or B = B2 ) ) )
A41: l1 <> l2 by A16, A17, A31;
 ( for i being Element of Seg d holds l1 . i <= r1 . i or for i being Element of Seg d holds r1 . i < l1 . i ) by A20;
hence  B1 <> B2 by A41, Th31; ::_thesis: for B being set holds
( not B in star A or B = B1 or B = B2 )
let B be set ; ::_thesis: ( not B in star A or B = B1 or B = B2 )
assume A42: B in star A ; ::_thesis: ( B = B1 or B = B2 )
then reconsider B = B as Cell of d,G by A1;
consider l9, r9 being Element of REAL d such that
A43: B = cell (l9,r9) and
A44: for i being Element of Seg d holds [(l9 . i),(r9 . i)] is Gap of G . i and
A45: ( for i being Element of Seg d holds l9 . i < r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by Th39;
A46: [(l9 . i0),(r9 . i0)] is Gap of G . i0 by A44;
A47: A c= B by A42, Th50;
percases ( for i being Element of Seg d holds l9 . i < r9 . i or for i being Element of Seg d holds r9 . i < l9 . i ) by A45;
supposeA48: for i being Element of Seg d holds l9 . i < r9 . i ; ::_thesis: ( B = B1 or B = B2 )
A49: now__::_thesis:_for_i_being_Element_of_Seg_d_st_i_<>_i0_holds_
(_l9_._i_=_l_._i_&_r9_._i_=_r_._i_)
let i be Element of Seg d; ::_thesis: ( i <> i0 implies ( l9 . i = l . i & r9 . i = r . i ) )
assume A50: i <> i0 ; ::_thesis: ( l9 . i = l . i & r9 . i = r . i )
 l9 . i < r9 . i by A48;
then  ( ( l . i = l9 . i & r . i = r9 . i ) or ( l . i = l9 . i & r . i = l9 . i ) or ( l . i = r9 . i & r . i = r9 . i ) ) by A2, A3, A43, A47, Th45;
hence  ( l9 . i = l . i & r9 . i = r . i ) by A6, A50; ::_thesis: verum
end;
thus  ( B = B1 or B = B2 ) ::_thesis: verum
proof
A51: l9 . i0 < r9 . i0 by A48;
percases ( ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) or ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) ) by A2, A3, A4, A43, A47, A51, Th45;
supposeA52: ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) ; ::_thesis: ( B = B1 or B = B2 )
then A53: l9 . i0 = l1 . i0 by A15, A16, A46, Th21;
reconsider l9 = l9, r9 = r9, l1 = l1, r1 = r1 as Function of (Seg d),REAL by Def3;
A54: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l9_._i_=_l1_._i_&_r9_._i_=_r1_._i_)
let i be Element of Seg d; ::_thesis: ( l9 . i = l1 . i & r9 . i = r1 . i )
A55: l1 . i0 < l . i0 by A48, A52, A53;
then  ( i = i0 or ( i <> i0 & l9 . i = l . i & l1 . i = l . i ) ) by A16, A17, A49;
hence  l9 . i = l1 . i by A15, A16, A46, A52, Th21; ::_thesis: r9 . i = r1 . i
 ( i = i0 or ( i <> i0 & r9 . i = r . i & r1 . i = r . i ) ) by A16, A17, A49, A55;
hence  r9 . i = r1 . i by A16, A52; ::_thesis: verum
end;
then  l9 = l1 by FUNCT_2:63;
hence  ( B = B1 or B = B2 ) by A43, A54, FUNCT_2:63; ::_thesis: verum
end;
supposeA56: ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) ; ::_thesis: ( B = B1 or B = B2 )
then A57: r9 . i0 = r2 . i0 by A30, A31, A46, Th20;
reconsider l9 = l9, r9 = r9, l2 = l2, r2 = r2 as Function of (Seg d),REAL by Def3;
A58: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_r9_._i_=_r2_._i_&_l9_._i_=_l2_._i_)
let i be Element of Seg d; ::_thesis: ( r9 . i = r2 . i & l9 . i = l2 . i )
A59: l . i0 < r2 . i0 by A48, A56, A57;
then  ( i = i0 or ( i <> i0 & r9 . i = r . i & r2 . i = r . i ) ) by A31, A32, A49;
hence  r9 . i = r2 . i by A30, A31, A46, A56, Th20; ::_thesis: l9 . i = l2 . i
 ( i = i0 or ( i <> i0 & l9 . i = l . i & l2 . i = l . i ) ) by A31, A32, A49, A59;
hence  l9 . i = l2 . i by A31, A56; ::_thesis: verum
end;
then  l9 = l2 by FUNCT_2:63;
hence  ( B = B1 or B = B2 ) by A43, A58, FUNCT_2:63; ::_thesis: verum
end;
end;
end;
end;
supposeA60: for i being Element of Seg d holds r9 . i < l9 . i ; ::_thesis: ( B = B1 or B = B2 )
consider i1 being Element of Seg d such that
A61: ( ( l . i1 = l9 . i1 & r . i1 = l9 . i1 ) or ( l . i1 = r9 . i1 & r . i1 = r9 . i1 ) ) by A2, A3, A43, A47, Th46;
A62: i0 = i1 by A6, A61;
thus  ( B = B1 or B = B2 ) ::_thesis: verum
proof
percases ( ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) or ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) ) by A61, A62;
supposeA63: ( l . i0 = r9 . i0 & r . i0 = r9 . i0 ) ; ::_thesis: ( B = B1 or B = B2 )
then  l9 . i0 = l1 . i0 by A15, A16, A46, Th21;
then  B1 = infinite-cell G by A16, A17, A60, A63, Th48;
hence  ( B = B1 or B = B2 ) by A43, A60, Th48; ::_thesis: verum
end;
supposeA64: ( l . i0 = l9 . i0 & r . i0 = l9 . i0 ) ; ::_thesis: ( B = B1 or B = B2 )
then  r9 . i0 = r2 . i0 by A30, A31, A46, Th20;
then  B2 = infinite-cell G by A31, A32, A60, A64, Th48;
hence  ( B = B1 or B = B2 ) by A43, A60, Th48; ::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  card (star A) = 2 by Th5; ::_thesis: verum
end;

theorem Th55: :: CHAIN_1:55
for d being non zero Element of NAT
for G being Grating of d
for B being Cell of (0 + 1),G holds card (del {B}) = 2
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for B being Cell of (0 + 1),G holds card (del {B}) = 2
A1: 0 + 1 <= d by Def2;
let G be Grating of d; ::_thesis: for B being Cell of (0 + 1),G holds card (del {B}) = 2
let B be Cell of (0 + 1),G; ::_thesis: card (del {B}) = 2
consider l, r being Element of REAL d, i0 being Element of Seg d such that
A2: B = cell (l,r) and
A3: ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) and
A4: [(l . i0),(r . i0)] is Gap of G . i0 and
A5: for i being Element of Seg d st i <> i0 holds
( l . i = r . i & l . i in G . i ) by Th43;
 ex A1, A2 being set st
( A1 in del {B} & A2 in del {B} & A1 <> A2 & ( for A being set holds
( not A in del {B} or A = A1 or A = A2 ) ) )
proof
 for i being Element of Seg d holds
( l . i in G . i & r . i in G . i ) by A1, A2, Th35;
then reconsider A1 = cell (l,l), A2 = cell (r,r) as Cell of 0,G by Th38;
take  A1 ; ::_thesis: ex A2 being set st
( A1 in del {B} & A2 in del {B} & A1 <> A2 & ( for A being set holds
( not A in del {B} or A = A1 or A = A2 ) ) )
take  A2 ; ::_thesis: ( A1 in del {B} & A2 in del {B} & A1 <> A2 & ( for A being set holds
( not A in del {B} or A = A1 or A = A2 ) ) )
A6: A1 = {l} by Th27;
A7: A2 = {r} by Th27;
A8: l in B by A2, Th26;
A9: r in B by A2, Th26;
A10: {l} c= B by A8, ZFMISC_1:31;
 {r} c= B by A9, ZFMISC_1:31;
hence  ( A1 in del {B} & A2 in del {B} ) by A1, A6, A7, A10, Th53; ::_thesis: ( A1 <> A2 & ( for A being set holds
( not A in del {B} or A = A1 or A = A2 ) ) )
thus  A1 <> A2 by A3, A6, A7, ZFMISC_1:3; ::_thesis: for A being set holds
( not A in del {B} or A = A1 or A = A2 )
let A be set ; ::_thesis: ( not A in del {B} or A = A1 or A = A2 )
assume A11: A in del {B} ; ::_thesis: ( A = A1 or A = A2 )
then reconsider A = A as Cell of 0,G ;
A12: A c= B by A1, A11, Th53;
consider x being Element of REAL d such that
A13: A = cell (x,x) and
A14: for i being Element of Seg d holds x . i in G . i by Th37;
A15: x in A by A13, Th26;
percases ( l . i0 < r . i0 or ( d = 1 & r . i0 < l . i0 ) ) by A3;
supposeA16: l . i0 < r . i0 ; ::_thesis: ( A = A1 or A = A2 )
A17: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_l_._i_<=_r_._i
let i be Element of Seg d; ::_thesis: l . i <= r . i
 ( i = i0 or i <> i0 ) ;
hence  l . i <= r . i by A5, A16; ::_thesis: verum
end;
A18: x . i0 in G . i0 by A14;
A19: l . i0 <= x . i0 by A2, A12, A15, A17, Th24;
A20: x . i0 <= r . i0 by A2, A12, A15, A17, Th24;
A21: ( not l . i0 < x . i0 or not x . i0 < r . i0 ) by A4, A18, Th16;
A22: now__::_thesis:_for_i_being_Element_of_Seg_d_st_i_<>_i0_holds_
(_x_._i_=_l_._i_&_x_._i_=_r_._i_)
let i be Element of Seg d; ::_thesis: ( i <> i0 implies ( x . i = l . i & x . i = r . i ) )
assume  i <> i0 ; ::_thesis: ( x . i = l . i & x . i = r . i )
then A23: l . i = r . i by A5;
A24: l . i <= x . i by A2, A12, A15, A17, Th24;
 x . i <= r . i by A2, A12, A15, A17, Th24;
hence  ( x . i = l . i & x . i = r . i ) by A23, A24, XXREAL_0:1; ::_thesis: verum
end;
thus  ( A = A1 or A = A2 ) ::_thesis: verum
proof
percases ( x . i0 = l . i0 or x . i0 = r . i0 ) by A19, A20, A21, XXREAL_0:1;
supposeA25: x . i0 = l . i0 ; ::_thesis: ( A = A1 or A = A2 )
reconsider x = x, l = l as Function of (Seg d),REAL by Def3;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_x_._i_=_l_._i
let i be Element of Seg d; ::_thesis: x . i = l . i
 ( i = i0 or i <> i0 ) ;
hence  x . i = l . i by A22, A25; ::_thesis: verum
end;
then  x = l by FUNCT_2:63;
hence  ( A = A1 or A = A2 ) by A13; ::_thesis: verum
end;
supposeA26: x . i0 = r . i0 ; ::_thesis: ( A = A1 or A = A2 )
reconsider x = x, r = r as Function of (Seg d),REAL by Def3;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_x_._i_=_r_._i
let i be Element of Seg d; ::_thesis: x . i = r . i
 ( i = i0 or i <> i0 ) ;
hence  x . i = r . i by A22, A26; ::_thesis: verum
end;
then  x = r by FUNCT_2:63;
hence  ( A = A1 or A = A2 ) by A13; ::_thesis: verum
end;
end;
end;
end;
supposeA27: ( d = 1 & r . i0 < l . i0 ) ; ::_thesis: ( A = A1 or A = A2 )
A28: for i being Element of Seg d holds i = i0
proof
let i be Element of Seg d; ::_thesis: i = i0
A29: 1 <= i by FINSEQ_1:1;
A30: i <= d by FINSEQ_1:1;
A31: 1 <= i0 by FINSEQ_1:1;
A32: i0 <= d by FINSEQ_1:1;
 i = 1 by A27, A29, A30, XXREAL_0:1;
hence  i = i0 by A27, A31, A32, XXREAL_0:1; ::_thesis: verum
end;
consider i1 being Element of Seg d such that
 r . i1 < l . i1 and
A33: ( x . i1 <= r . i1 or l . i1 <= x . i1 ) by A2, A12, A15, A27, Th25;
A34: i1 = i0 by A28;
A35: x . i0 in G . i0 by A14;
then A36: not x . i0 < r . i0 by A4, A27, Th16;
A37: not l . i0 < x . i0 by A4, A27, A35, Th16;
thus  ( A = A1 or A = A2 ) ::_thesis: verum
proof
percases ( x . i0 = r . i0 or x . i0 = l . i0 ) by A33, A34, A36, A37, XXREAL_0:1;
supposeA38: x . i0 = r . i0 ; ::_thesis: ( A = A1 or A = A2 )
reconsider x = x, r = r as Function of (Seg d),REAL by Def3;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_x_._i_=_r_._i
let i be Element of Seg d; ::_thesis: x . i = r . i
 i = i0 by A28;
hence  x . i = r . i by A38; ::_thesis: verum
end;
then  x = r by FUNCT_2:63;
hence  ( A = A1 or A = A2 ) by A13; ::_thesis: verum
end;
supposeA39: x . i0 = l . i0 ; ::_thesis: ( A = A1 or A = A2 )
reconsider x = x, l = l as Function of (Seg d),REAL by Def3;
now__::_thesis:_for_i_being_Element_of_Seg_d_holds_x_._i_=_l_._i
let i be Element of Seg d; ::_thesis: x . i = l . i
 i = i0 by A28;
hence  x . i = l . i by A39; ::_thesis: verum
end;
then  x = l by FUNCT_2:63;
hence  ( A = A1 or A = A2 ) by A13; ::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  card (del {B}) = 2 by Th5; ::_thesis: verum
end;

theorem :: CHAIN_1:56
for d being non zero Element of NAT
for G being Grating of d holds
( Omega G = (0_ (d,G)) ` & 0_ (d,G) = (Omega G) ` )
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d holds
( Omega G = (0_ (d,G)) ` & 0_ (d,G) = (Omega G) ` )
let G be Grating of d; ::_thesis: ( Omega G = (0_ (d,G)) ` & 0_ (d,G) = (Omega G) ` )
 Omega G = (0_ (d,G)) ` ;
hence  ( Omega G = (0_ (d,G)) ` & 0_ (d,G) = (Omega G) ` ) ; ::_thesis: verum
end;

theorem :: CHAIN_1:57
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for C being Chain of k,G holds C + (0_ (k,G)) = C ;

theorem Th58: :: CHAIN_1:58
for d being non zero Element of NAT
for G being Grating of d
for C being Chain of d,G holds C ` = C + (Omega G)
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C being Chain of d,G holds C ` = C + (Omega G)
let G be Grating of d; ::_thesis: for C being Chain of d,G holds C ` = C + (Omega G)
let C be Chain of d,G; ::_thesis: C ` = C + (Omega G)
 C /\ (cells (d,G)) = C by XBOOLE_1:28;
hence  C ` = C + (Omega G) by XBOOLE_1:100; ::_thesis: verum
end;

theorem Th59: :: CHAIN_1:59
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d holds del (0_ ((k + 1),G)) = 0_ (k,G)
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d holds del (0_ ((k + 1),G)) = 0_ (k,G)
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d holds del (0_ ((k + 1),G)) = 0_ (k,G)
let G be Grating of d; ::_thesis: del (0_ ((k + 1),G)) = 0_ (k,G)
now__::_thesis:_for_A_being_Cell_of_k,G_holds_
(_A_in_del_(0__((k_+_1),G))_iff_A_in_0__(k,G)_)
let A be Cell of k,G; ::_thesis: ( A in del (0_ ((k + 1),G)) iff A in 0_ (k,G) )
 card ((star A) /\ (0_ ((k + 1),G))) = 2 * 0 ;
hence  ( A in del (0_ ((k + 1),G)) iff A in 0_ (k,G) ) by Th51; ::_thesis: verum
end;
hence  del (0_ ((k + 1),G)) = 0_ (k,G) by SUBSET_1:3; ::_thesis: verum
end;

theorem Th60: :: CHAIN_1:60
for d9 being Element of NAT
for G being Grating of d9 + 1 holds del (Omega G) = 0_ (d9,G)
proof
let d9 be Element of NAT ; ::_thesis: for G being Grating of d9 + 1 holds del (Omega G) = 0_ (d9,G)
let G be Grating of d9 + 1; ::_thesis: del (Omega G) = 0_ (d9,G)
now__::_thesis:_for_A_being_Cell_of_d9,G_holds_
(_A_in_del_(Omega_G)_iff_A_in_0__(d9,G)_)
let A be Cell of d9,G; ::_thesis: ( A in del (Omega G) iff A in 0_ (d9,G) )
 (star A) /\ (Omega G) = star A by XBOOLE_1:28;
then  card ((star A) /\ (Omega G)) = 2 * 1 by Th54;
hence  ( A in del (Omega G) iff A in 0_ (d9,G) ) by Th51; ::_thesis: verum
end;
hence  del (Omega G) = 0_ (d9,G) by SUBSET_1:3; ::_thesis: verum
end;

theorem Th61: :: CHAIN_1:61
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for C1, C2 being Chain of (k + 1),G holds del (C1 + C2) = (del C1) + (del C2)
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d
for C1, C2 being Chain of (k + 1),G holds del (C1 + C2) = (del C1) + (del C2)
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C1, C2 being Chain of (k + 1),G holds del (C1 + C2) = (del C1) + (del C2)
let G be Grating of d; ::_thesis: for C1, C2 being Chain of (k + 1),G holds del (C1 + C2) = (del C1) + (del C2)
let C1, C2 be Chain of (k + 1),G; ::_thesis: del (C1 + C2) = (del C1) + (del C2)
now__::_thesis:_for_A_being_Cell_of_k,G_holds_
(_A_in_del_(C1_+_C2)_iff_A_in_(del_C1)_+_(del_C2)_)
let A be Cell of k,G; ::_thesis: ( A in del (C1 + C2) iff A in (del C1) + (del C2) )
A1: (star A) /\ (C1 \+\ C2) = ((star A) /\ C1) \+\ ((star A) /\ C2) by XBOOLE_1:112;
A2: ( A in del (C1 + C2) iff ( k + 1 <= d & card ((star A) /\ (C1 \+\ C2)) is odd ) ) by Th51;
A3: ( A in del C1 iff ( k + 1 <= d & card ((star A) /\ C1) is odd ) ) by Th51;
 ( A in del C2 iff ( k + 1 <= d & card ((star A) /\ C2) is odd ) ) by Th51;
hence  ( A in del (C1 + C2) iff A in (del C1) + (del C2) ) by A1, A2, A3, Th8, XBOOLE_0:1; ::_thesis: verum
end;
hence  del (C1 + C2) = (del C1) + (del C2) by SUBSET_1:3; ::_thesis: verum
end;

theorem Th62: :: CHAIN_1:62
for d9 being Element of NAT
for G being Grating of d9 + 1
for C being Chain of (d9 + 1),G holds del (C `) = del C
proof
let d9 be Element of NAT ; ::_thesis: for G being Grating of d9 + 1
for C being Chain of (d9 + 1),G holds del (C `) = del C
let G be Grating of d9 + 1; ::_thesis: for C being Chain of (d9 + 1),G holds del (C `) = del C
let C be Chain of (d9 + 1),G; ::_thesis: del (C `) = del C
thus  del (C `) = del (C + (Omega G)) by Th58
.= (del C) + (del (Omega G)) by Th61
.= (del C) + (0_ (d9,G)) by Th60
.= del C ; ::_thesis: verum
end;

theorem Th63: :: CHAIN_1:63
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for C being Chain of ((k + 1) + 1),G holds del (del C) = 0_ (k,G)
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d
for C being Chain of ((k + 1) + 1),G holds del (del C) = 0_ (k,G)
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C being Chain of ((k + 1) + 1),G holds del (del C) = 0_ (k,G)
let G be Grating of d; ::_thesis: for C being Chain of ((k + 1) + 1),G holds del (del C) = 0_ (k,G)
let C be Chain of ((k + 1) + 1),G; ::_thesis: del (del C) = 0_ (k,G)
percases ( (k + 1) + 1 <= d or (k + 1) + 1 > d ) ;
supposeA1: (k + 1) + 1 <= d ; ::_thesis: del (del C) = 0_ (k,G)
then A2: k + 1 < d by NAT_1:13;
then A3: k < d by NAT_1:13;
A4: for C being Cell of ((k + 1) + 1),G
for l, r being Element of REAL d st C = cell (l,r) & ( for i being Element of Seg d holds l . i <= r . i ) holds
del (del {C}) = 0_ (k,G)
proof
let C be Cell of ((k + 1) + 1),G; ::_thesis: for l, r being Element of REAL d st C = cell (l,r) & ( for i being Element of Seg d holds l . i <= r . i ) holds
del (del {C}) = 0_ (k,G)
let l, r be Element of REAL d; ::_thesis: ( C = cell (l,r) & ( for i being Element of Seg d holds l . i <= r . i ) implies del (del {C}) = 0_ (k,G) )
assume that
A5: C = cell (l,r) and
A6: for i being Element of Seg d holds l . i <= r . i ; ::_thesis: del (del {C}) = 0_ (k,G)
now__::_thesis:_for_A_being_set_holds_not_A_in_del_(del_{C})
let A be set ; ::_thesis: not A in del (del {C})
assume A7: A in del (del {C}) ; ::_thesis: contradiction
then reconsider A = A as Cell of k,G ;
set BB = (star A) /\ (del {C});
A8: now__::_thesis:_for_B_being_Cell_of_(k_+_1),G_holds_
(_B_in_(star_A)_/\_(del_{C})_iff_(_A_c=_B_&_B_c=_C_)_)
let B be Cell of (k + 1),G; ::_thesis: ( B in (star A) /\ (del {C}) iff ( A c= B & B c= C ) )
 ( B in (star A) /\ (del {C}) iff ( B in star A & B in del {C} ) ) by XBOOLE_0:def_4;
hence  ( B in (star A) /\ (del {C}) iff ( A c= B & B c= C ) ) by A1, Th50, Th53; ::_thesis: verum
end;
A9: card ((star A) /\ (del {C})) is odd by A7, Th51;
consider B being set  such that
A10: B in (star A) /\ (del {C}) by A9, CARD_1:27, XBOOLE_0:def_1;
reconsider B = B as Cell of (k + 1),G by A10;
A11: A c= B by A8, A10;
 B c= C by A8, A10;
then A12: A c= C by A11, XBOOLE_1:1;
set i0 = the Element of Seg d;
 l . the Element of Seg d <= r . the Element of Seg d by A6;
then consider Z being Subset of (Seg d) such that
A13: card Z = (k + 1) + 1 and
A14: for i being Element of Seg d holds
( ( i in Z & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in Z & l . i = r . i & l . i in G . i ) ) by A1, A5, Th33;
consider l9, r9 being Element of REAL d such that
A15: A = cell (l9,r9) and
A16: ( ex X being Subset of (Seg d) st
( card X = k & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or ( k = d & ( for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) ) ) by A3, Th32;
 l9 . the Element of Seg d <= r9 . the Element of Seg d by A5, A6, A12, A15, Th28;
then consider X being Subset of (Seg d) such that
A17: card X = k and
A18: for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) by A16;
 ex B1, B2 being set st
( B1 in (star A) /\ (del {C}) & B2 in (star A) /\ (del {C}) & B1 <> B2 & ( for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 ) ) )
proof
A19: X c= Z by A5, A12, A14, A15, A18, Th47;
then  card (Z \ X) = (k + (1 + 1)) - k by A13, A17, CARD_2:44
.= 2 ;
then consider i1, i2 being set  such that
A20: i1 in Z \ X and
A21: i2 in Z \ X and
A22: i1 <> i2 and
A23: for i being set holds
( not i in Z \ X or i = i1 or i = i2 ) by Th5;
A24: i1 in Z by A20, XBOOLE_0:def_5;
A25: i2 in Z by A21, XBOOLE_0:def_5;
A26: not i1 in X by A20, XBOOLE_0:def_5;
A27: not i2 in X by A21, XBOOLE_0:def_5;
reconsider i1 = i1, i2 = i2 as Element of Seg d by A20, A21;
set Y1 = X \/ {i1};
A28: X c= X \/ {i1} by XBOOLE_1:7;
 {i1} c= Z by A24, ZFMISC_1:31;
then A29: X \/ {i1} c= Z by A19, XBOOLE_1:8;
defpred S1[ Element of Seg d, Real] means ( ( $1 in X \/ {i1} implies $2 = l . $1 ) & ( not $1 in X \/ {i1} implies $2 = l9 . $1 ) );
A30: for i being Element of Seg d ex xi being Real st S1[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S1[i,xi]
 ( i in X \/ {i1} or not i in X \/ {i1} ) ;
hence  ex xi being Real st S1[i,xi] ; ::_thesis: verum
end;
consider l1 being Function of (Seg d),REAL such that
A31: for i being Element of Seg d holds S1[i,l1 . i] from FUNCT_2:sch_3(A30);
defpred S2[ Element of Seg d, Real] means ( ( $1 in X \/ {i1} implies $2 = r . $1 ) & ( not $1 in X \/ {i1} implies $2 = r9 . $1 ) );
A32: for i being Element of Seg d ex xi being Real st S2[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S2[i,xi]
 ( i in X \/ {i1} or not i in X \/ {i1} ) ;
hence  ex xi being Real st S2[i,xi] ; ::_thesis: verum
end;
consider r1 being Function of (Seg d),REAL such that
A33: for i being Element of Seg d holds S2[i,r1 . i] from FUNCT_2:sch_3(A32);
reconsider l1 = l1, r1 = r1 as Element of REAL d by Def3;
A34: for i being Element of Seg d holds l1 . i <= r1 . i
proof
let i be Element of Seg d; ::_thesis: l1 . i <= r1 . i
 ( ( l1 . i = l . i & r1 . i = r . i ) or ( l1 . i = l9 . i & r1 . i = r9 . i ) ) by A31, A33;
hence  l1 . i <= r1 . i by A14, A18; ::_thesis: verum
end;
A35: card (X \/ {i1}) = (card X) + (card {i1}) by A26, CARD_2:40, ZFMISC_1:50
.= k + 1 by A17, CARD_1:30 ;
 for i being Element of Seg d holds
( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) )
proof
let i be Element of Seg d; ::_thesis: ( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) )
percases ( i in X \/ {i1} or not i in X \/ {i1} ) ;
supposeA36: i in X \/ {i1} ; ::_thesis: ( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) )
then A37: l1 . i = l . i by A31;
 r1 . i = r . i by A33, A36;
hence  ( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) ) by A14, A29, A36, A37; ::_thesis: verum
end;
supposeA38: not i in X \/ {i1} ; ::_thesis: ( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) )
then A39: l1 . i = l9 . i by A31;
A40: r1 . i = r9 . i by A33, A38;
 not i in X by A28, A38;
hence  ( ( i in X \/ {i1} & l1 . i < r1 . i & [(l1 . i),(r1 . i)] is Gap of G . i ) or ( not i in X \/ {i1} & l1 . i = r1 . i & l1 . i in G . i ) ) by A18, A38, A39, A40; ::_thesis: verum
end;
end;
end;
then reconsider B1 = cell (l1,r1) as Cell of (k + 1),G by A2, A35, Th33;
set Y2 = X \/ {i2};
A41: X c= X \/ {i2} by XBOOLE_1:7;
 {i2} c= Z by A25, ZFMISC_1:31;
then A42: X \/ {i2} c= Z by A19, XBOOLE_1:8;
defpred S3[ Element of Seg d, Real] means ( ( $1 in X \/ {i2} implies $2 = l . $1 ) & ( not $1 in X \/ {i2} implies $2 = l9 . $1 ) );
A43: for i being Element of Seg d ex xi being Real st S3[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S3[i,xi]
 ( i in X \/ {i2} or not i in X \/ {i2} ) ;
hence  ex xi being Real st S3[i,xi] ; ::_thesis: verum
end;
consider l2 being Function of (Seg d),REAL such that
A44: for i being Element of Seg d holds S3[i,l2 . i] from FUNCT_2:sch_3(A43);
defpred S4[ Element of Seg d, Real] means ( ( $1 in X \/ {i2} implies $2 = r . $1 ) & ( not $1 in X \/ {i2} implies $2 = r9 . $1 ) );
A45: for i being Element of Seg d ex xi being Real st S4[i,xi]
proof
let i be Element of Seg d; ::_thesis: ex xi being Real st S4[i,xi]
 ( i in X \/ {i2} or not i in X \/ {i2} ) ;
hence  ex xi being Real st S4[i,xi] ; ::_thesis: verum
end;
consider r2 being Function of (Seg d),REAL such that
A46: for i being Element of Seg d holds S4[i,r2 . i] from FUNCT_2:sch_3(A45);
reconsider l2 = l2, r2 = r2 as Element of REAL d by Def3;
A47: card (X \/ {i2}) = (card X) + (card {i2}) by A27, CARD_2:40, ZFMISC_1:50
.= k + 1 by A17, CARD_1:30 ;
 for i being Element of Seg d holds
( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) )
proof
let i be Element of Seg d; ::_thesis: ( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) )
percases ( i in X \/ {i2} or not i in X \/ {i2} ) ;
supposeA48: i in X \/ {i2} ; ::_thesis: ( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) )
then A49: l2 . i = l . i by A44;
 r2 . i = r . i by A46, A48;
hence  ( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) ) by A14, A42, A48, A49; ::_thesis: verum
end;
supposeA50: not i in X \/ {i2} ; ::_thesis: ( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) )
then A51: l2 . i = l9 . i by A44;
A52: r2 . i = r9 . i by A46, A50;
 not i in X by A41, A50;
hence  ( ( i in X \/ {i2} & l2 . i < r2 . i & [(l2 . i),(r2 . i)] is Gap of G . i ) or ( not i in X \/ {i2} & l2 . i = r2 . i & l2 . i in G . i ) ) by A18, A50, A51, A52; ::_thesis: verum
end;
end;
end;
then reconsider B2 = cell (l2,r2) as Cell of (k + 1),G by A2, A47, Th33;
take  B1 ; ::_thesis: ex B2 being set st
( B1 in (star A) /\ (del {C}) & B2 in (star A) /\ (del {C}) & B1 <> B2 & ( for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 ) ) )
take  B2 ; ::_thesis: ( B1 in (star A) /\ (del {C}) & B2 in (star A) /\ (del {C}) & B1 <> B2 & ( for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 ) ) )
A53: for i being Element of Seg d holds
( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i )
proof
let i be Element of Seg d; ::_thesis: ( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i )
percases ( i in X \/ {i1} or not i in X \/ {i1} ) ;
supposeA54: i in X \/ {i1} ; ::_thesis: ( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i )
then A55: l1 . i = l . i by A31;
 r1 . i = r . i by A33, A54;
hence  ( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i ) by A5, A6, A12, A15, A55, Th28; ::_thesis: verum
end;
supposeA56: not i in X \/ {i1} ; ::_thesis: ( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i )
then A57: l1 . i = l9 . i by A31;
 r1 . i = r9 . i by A33, A56;
hence  ( l1 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r1 . i & l . i <= l1 . i & l1 . i <= r1 . i & r1 . i <= r . i ) by A5, A6, A12, A15, A57, Th28; ::_thesis: verum
end;
end;
end;
then A58: A c= B1 by A15, Th28;
 B1 c= C by A5, A6, A53, Th28;
hence  B1 in (star A) /\ (del {C}) by A8, A58; ::_thesis: ( B2 in (star A) /\ (del {C}) & B1 <> B2 & ( for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 ) ) )
A59: for i being Element of Seg d holds
( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i )
proof
let i be Element of Seg d; ::_thesis: ( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i )
percases ( i in X \/ {i2} or not i in X \/ {i2} ) ;
supposeA60: i in X \/ {i2} ; ::_thesis: ( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i )
then A61: l2 . i = l . i by A44;
 r2 . i = r . i by A46, A60;
hence  ( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i ) by A5, A6, A12, A15, A61, Th28; ::_thesis: verum
end;
supposeA62: not i in X \/ {i2} ; ::_thesis: ( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i )
then A63: l2 . i = l9 . i by A44;
 r2 . i = r9 . i by A46, A62;
hence  ( l2 . i <= l9 . i & l9 . i <= r9 . i & r9 . i <= r2 . i & l . i <= l2 . i & l2 . i <= r2 . i & r2 . i <= r . i ) by A5, A6, A12, A15, A63, Th28; ::_thesis: verum
end;
end;
end;
then A64: A c= B2 by A15, Th28;
 B2 c= C by A5, A6, A59, Th28;
hence  B2 in (star A) /\ (del {C}) by A8, A64; ::_thesis: ( B1 <> B2 & ( for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 ) ) )
 i1 in {i1} by TARSKI:def_1;
then A65: i1 in X \/ {i1} by XBOOLE_0:def_3;
A66: not i1 in X by A20, XBOOLE_0:def_5;
 not i1 in {i2} by A22, TARSKI:def_1;
then A67: not i1 in X \/ {i2} by A66, XBOOLE_0:def_3;
A68: l1 . i1 = l . i1 by A31, A65;
A69: r1 . i1 = r . i1 by A33, A65;
A70: l2 . i1 = l9 . i1 by A44, A67;
A71: r2 . i1 = r9 . i1 by A46, A67;
 l . i1 < r . i1 by A14, A24;
then  ( l1 <> l2 or r1 <> r2 ) by A18, A26, A68, A69, A70, A71;
hence  B1 <> B2 by A34, Th31; ::_thesis: for B being set holds
( not B in (star A) /\ (del {C}) or B = B1 or B = B2 )
let B be set ; ::_thesis: ( not B in (star A) /\ (del {C}) or B = B1 or B = B2 )
assume A72: B in (star A) /\ (del {C}) ; ::_thesis: ( B = B1 or B = B2 )
then reconsider B = B as Cell of (k + 1),G ;
A73: A c= B by A8, A72;
A74: B c= C by A8, A72;
consider l99, r99 being Element of REAL d such that
A75: B = cell (l99,r99) and
A76: ( ex Y being Subset of (Seg d) st
( card Y = k + 1 & ( for i being Element of Seg d holds
( ( i in Y & l99 . i < r99 . i & [(l99 . i),(r99 . i)] is Gap of G . i ) or ( not i in Y & l99 . i = r99 . i & l99 . i in G . i ) ) ) ) or ( k + 1 = d & ( for i being Element of Seg d holds
( r99 . i < l99 . i & [(l99 . i),(r99 . i)] is Gap of G . i ) ) ) ) by A2, Th32;
 l99 . the Element of Seg d <= r99 . the Element of Seg d by A5, A6, A74, A75, Th28;
then consider Y being Subset of (Seg d) such that
A77: card Y = k + 1 and
A78: for i being Element of Seg d holds
( ( i in Y & l99 . i < r99 . i & [(l99 . i),(r99 . i)] is Gap of G . i ) or ( not i in Y & l99 . i = r99 . i & l99 . i in G . i ) ) by A76;
A79: X c= Y by A15, A18, A73, A75, A78, Th47;
A80: Y c= Z by A5, A14, A74, A75, A78, Th47;
card (Y \ X) = (k + 1) - k by A17, A77, A79, CARD_2:44
.= 1 ;
then consider i9 being set  such that
A81: Y \ X = {i9} by CARD_2:42;
A82: i9 in Y \ X by A81, TARSKI:def_1;
then reconsider i9 = i9 as Element of Seg d ;
A83: i9 in Y by A82, XBOOLE_0:def_5;
 not i9 in X by A82, XBOOLE_0:def_5;
then A84: i9 in Z \ X by A80, A83, XBOOLE_0:def_5;
A85: Y = X \/ Y by A79, XBOOLE_1:12
.= X \/ {i9} by A81, XBOOLE_1:39 ;
percases ( Y = X \/ {i1} or Y = X \/ {i2} ) by A23, A84, A85;
supposeA86: Y = X \/ {i1} ; ::_thesis: ( B = B1 or B = B2 )
reconsider l99 = l99, r99 = r99, l1 = l1, r1 = r1 as Function of (Seg d),REAL by Def3;
A87: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l99_._i_=_l1_._i_&_r99_._i_=_r1_._i_)
let i be Element of Seg d; ::_thesis: ( l99 . i = l1 . i & r99 . i = r1 . i )
 ( i in Y or not i in Y ) ;
then  ( ( l99 . i = l . i & l1 . i = l . i & r99 . i = r . i & r1 . i = r . i ) or ( l99 . i = l9 . i & l1 . i = l9 . i & r99 . i = r9 . i & r1 . i = r9 . i ) ) by A5, A14, A15, A18, A31, A33, A73, A74, A75, A78, A86, Th47;
hence  ( l99 . i = l1 . i & r99 . i = r1 . i ) ; ::_thesis: verum
end;
then  l99 = l1 by FUNCT_2:63;
hence  ( B = B1 or B = B2 ) by A75, A87, FUNCT_2:63; ::_thesis: verum
end;
supposeA88: Y = X \/ {i2} ; ::_thesis: ( B = B1 or B = B2 )
reconsider l99 = l99, r99 = r99, l2 = l2, r2 = r2 as Function of (Seg d),REAL by Def3;
A89: now__::_thesis:_for_i_being_Element_of_Seg_d_holds_
(_l99_._i_=_l2_._i_&_r99_._i_=_r2_._i_)
let i be Element of Seg d; ::_thesis: ( l99 . i = l2 . i & r99 . i = r2 . i )
 ( i in Y or not i in Y ) ;
then  ( ( l99 . i = l . i & l2 . i = l . i & r99 . i = r . i & r2 . i = r . i ) or ( l99 . i = l9 . i & l2 . i = l9 . i & r99 . i = r9 . i & r2 . i = r9 . i ) ) by A5, A14, A15, A18, A44, A46, A73, A74, A75, A78, A88, Th47;
hence  ( l99 . i = l2 . i & r99 . i = r2 . i ) ; ::_thesis: verum
end;
then  l99 = l2 by FUNCT_2:63;
hence  ( B = B1 or B = B2 ) by A75, A89, FUNCT_2:63; ::_thesis: verum
end;
end;
end;
then  card ((star A) /\ (del {C})) = 2 * 1 by Th5;
hence  contradiction by A7, Th51; ::_thesis: verum
end;
hence  del (del {C}) = 0_ (k,G) by XBOOLE_0:def_1; ::_thesis: verum
end;
A90: for C1, C2 being Chain of ((k + 1) + 1),G st del (del C1) = 0_ (k,G) & del (del C2) = 0_ (k,G) holds
del (del (C1 + C2)) = 0_ (k,G)
proof
let C1, C2 be Chain of ((k + 1) + 1),G; ::_thesis: ( del (del C1) = 0_ (k,G) & del (del C2) = 0_ (k,G) implies del (del (C1 + C2)) = 0_ (k,G) )
assume that
A91: del (del C1) = 0_ (k,G) and
A92: del (del C2) = 0_ (k,G) ; ::_thesis: del (del (C1 + C2)) = 0_ (k,G)
thus  del (del (C1 + C2)) = del ((del C1) + (del C2)) by Th61
.= (0_ (k,G)) + (0_ (k,G)) by A91, A92, Th61
.= 0_ (k,G) ; ::_thesis: verum
end;
defpred S1[ Chain of ((k + 1) + 1),G] means  del (del $1) = 0_ (k,G);
del (del (0_ (((k + 1) + 1),G))) = del (0_ ((k + 1),G)) by Th59
.= 0_ (k,G) by Th59 ;
then A93: S1[ 0_ (((k + 1) + 1),G)] ;
 for A being Cell of ((k + 1) + 1),G holds del (del {A}) = 0_ (k,G)
proof
let A be Cell of ((k + 1) + 1),G; ::_thesis: del (del {A}) = 0_ (k,G)
consider l, r being Element of REAL d such that
A94: A = cell (l,r) and
A95: ( ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( (k + 1) + 1 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A1, Th32;
percases ( ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) or ( (k + 1) + 1 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ) by A95;
suppose ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l . i < r . i & [(l . i),(r . i)] is Gap of G . i ) or ( not i in X & l . i = r . i & l . i in G . i ) ) ) ) ; ::_thesis: del (del {A}) = 0_ (k,G)
then  for i being Element of Seg d holds l . i <= r . i ;
hence  del (del {A}) = 0_ (k,G) by A4, A94; ::_thesis: verum
end;
supposeA96: ( (k + 1) + 1 = d & ( for i being Element of Seg d holds
( r . i < l . i & [(l . i),(r . i)] is Gap of G . i ) ) ) ; ::_thesis: del (del {A}) = 0_ (k,G)
then A97: A = infinite-cell G by A94, Th49;
set C = {A} ` ;
A98: for A being Cell of ((k + 1) + 1),G st A in {A} ` holds
S1[{A}]
proof
let A9 be Cell of ((k + 1) + 1),G; ::_thesis: ( A9 in {A} ` implies S1[{A9}] )
assume  A9 in {A} ` ; ::_thesis: S1[{A9}]
then  not A9 in {A} by XBOOLE_0:def_5;
then A99: A9 <> infinite-cell G by A97, TARSKI:def_1;
consider l9, r9 being Element of REAL d such that
A100: A9 = cell (l9,r9) and
A101: ( ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or ( (k + 1) + 1 = d & ( for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) ) ) by A1, Th32;
percases ( ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) or for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ) by A101;
suppose ex X being Subset of (Seg d) st
( card X = (k + 1) + 1 & ( for i being Element of Seg d holds
( ( i in X & l9 . i < r9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) or ( not i in X & l9 . i = r9 . i & l9 . i in G . i ) ) ) ) ; ::_thesis: S1[{A9}]
then  for i being Element of Seg d holds l9 . i <= r9 . i ;
hence  S1[{A9}] by A4, A100; ::_thesis: verum
end;
suppose for i being Element of Seg d holds
( r9 . i < l9 . i & [(l9 . i),(r9 . i)] is Gap of G . i ) ; ::_thesis: S1[{A9}]
hence  S1[{A9}] by A99, A100, Th49; ::_thesis: verum
end;
end;
end;
A102: for C1, C2 being Chain of ((k + 1) + 1),G st C1 c= {A} ` & C2 c= {A} ` & S1[C1] & S1[C2] holds
S1[C1 + C2] by A90;
 S1[{A} ` ] from CHAIN_1:sch_4(A93, A98, A102);
hence  del (del {A}) = 0_ (k,G) by A96, Th62; ::_thesis: verum
end;
end;
end;
then A103: for A being Cell of ((k + 1) + 1),G st A in C holds
S1[{A}] ;
A104: for C1, C2 being Chain of ((k + 1) + 1),G st C1 c= C & C2 c= C & S1[C1] & S1[C2] holds
S1[C1 + C2] by A90;
thus  S1[C] from CHAIN_1:sch_4(A93, A103, A104); ::_thesis: verum
end;
suppose (k + 1) + 1 > d ; ::_thesis: del (del C) = 0_ (k,G)
then  del C = 0_ ((k + 1),G) by Th52;
hence  del (del C) = 0_ (k,G) by Th59; ::_thesis: verum
end;
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
mode Cycle of k,G ->  Chain of k,G means :Def14: :: CHAIN_1:def 14
( ( k = 0 & card it is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = it & del C = 0_ (k9,G) ) ) );
existence
 ex b1 being Chain of k,G st
( ( k = 0 & card b1 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = b1 & del C = 0_ (k9,G) ) ) )
proof
percases ( k = 0 or ex k9 being Nat st k = k9 + 1 ) by NAT_1:6;
supposeA1: k = 0 ; ::_thesis: ex b1 being Chain of k,G st
( ( k = 0 & card b1 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = b1 & del C = 0_ (k9,G) ) ) )
take  0_ (k,G) ; ::_thesis: ( ( k = 0 & card (0_ (k,G)) is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = 0_ (k,G) & del C = 0_ (k9,G) ) ) )
thus  ( ( k = 0 & card (0_ (k,G)) is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = 0_ (k,G) & del C = 0_ (k9,G) ) ) ) by A1; ::_thesis: verum
end;
suppose ex k9 being Nat st k = k9 + 1 ; ::_thesis: ex b1 being Chain of k,G st
( ( k = 0 & card b1 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = b1 & del C = 0_ (k9,G) ) ) )
then consider k9 being Nat such that
A2: k = k9 + 1 ;
reconsider k9 = k9 as Element of NAT by ORDINAL1:def_12;
take C9 = 0_ (k,G); ::_thesis: ( ( k = 0 & card C9 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = C9 & del C = 0_ (k9,G) ) ) )
now__::_thesis:_ex_k9_being_Element_of_NAT_st_
(_k_=_k9_+_1_&_ex_C_being_Chain_of_(k9_+_1),G_st_
(_C_=_C9_&_del_C_=_0__(k9,G)_)_)
take k9 = k9; ::_thesis: ( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = C9 & del C = 0_ (k9,G) ) )
thus  k = k9 + 1 by A2; ::_thesis: ex C being Chain of (k9 + 1),G st
( C = C9 & del C = 0_ (k9,G) )
reconsider C = C9 as Chain of (k9 + 1),G by A2;
take C = C; ::_thesis: ( C = C9 & del C = 0_ (k9,G) )
thus  ( C = C9 & del C = 0_ (k9,G) ) by A2, Th59; ::_thesis: verum
end;
hence  ( ( k = 0 & card C9 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = C9 & del C = 0_ (k9,G) ) ) ) ; ::_thesis: verum
end;
end;
end;
end;

:: deftheorem Def14 defines Cycle CHAIN_1:def_14_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for b4 being Chain of k,G holds
( b4 is Cycle of k,G iff ( ( k = 0 & card b4 is even ) or ex k9 being Element of NAT st
( k = k9 + 1 & ex C being Chain of (k9 + 1),G st
( C = b4 & del C = 0_ (k9,G) ) ) ) );

theorem Th64: :: CHAIN_1:64
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for C being Chain of (k + 1),G holds
( C is Cycle of k + 1,G iff del C = 0_ (k,G) )
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d
for C being Chain of (k + 1),G holds
( C is Cycle of k + 1,G iff del C = 0_ (k,G) )
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C being Chain of (k + 1),G holds
( C is Cycle of k + 1,G iff del C = 0_ (k,G) )
let G be Grating of d; ::_thesis: for C being Chain of (k + 1),G holds
( C is Cycle of k + 1,G iff del C = 0_ (k,G) )
let C be Chain of (k + 1),G; ::_thesis: ( C is Cycle of k + 1,G iff del C = 0_ (k,G) )
hereby  ::_thesis: ( del C = 0_ (k,G) implies C is Cycle of k + 1,G )
assume  C is Cycle of k + 1,G ; ::_thesis: del C = 0_ (k,G)
then  ex k9 being Element of NAT st
( k + 1 = k9 + 1 & ex C9 being Chain of (k9 + 1),G st
( C9 = C & del C9 = 0_ (k9,G) ) ) by Def14;
hence  del C = 0_ (k,G) ; ::_thesis: verum
end;
thus  ( del C = 0_ (k,G) implies C is Cycle of k + 1,G ) by Def14; ::_thesis: verum
end;

theorem :: CHAIN_1:65
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d st k > d holds
for C being Chain of k,G holds C is Cycle of k,G
proof
let k be Element of NAT ; ::_thesis: for d being non zero Element of NAT
for G being Grating of d st k > d holds
for C being Chain of k,G holds C is Cycle of k,G
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d st k > d holds
for C being Chain of k,G holds C is Cycle of k,G
let G be Grating of d; ::_thesis: ( k > d implies for C being Chain of k,G holds C is Cycle of k,G )
assume A1: k > d ; ::_thesis: for C being Chain of k,G holds C is Cycle of k,G
let C be Chain of k,G; ::_thesis: C is Cycle of k,G
consider k9 being Nat such that
A2: k = k9 + 1 by A1, NAT_1:6;
reconsider k9 = k9 as Element of NAT by ORDINAL1:def_12;
reconsider C9 = C as Chain of (k9 + 1),G by A2;
 del C9 = 0_ (k9,G) by A1, A2, Th52;
hence  C is Cycle of k,G by A2, Def14; ::_thesis: verum
end;

theorem Th66: :: CHAIN_1:66
for d being non zero Element of NAT
for G being Grating of d
for C being Chain of 0,G holds
( C is Cycle of 0 ,G iff card C is even )
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C being Chain of 0,G holds
( C is Cycle of 0 ,G iff card C is even )
let G be Grating of d; ::_thesis: for C being Chain of 0,G holds
( C is Cycle of 0 ,G iff card C is even )
let C be Chain of 0,G; ::_thesis: ( C is Cycle of 0 ,G iff card C is even )
hereby  ::_thesis: ( card C is even implies C is Cycle of 0 ,G )
assume  C is Cycle of 0 ,G ; ::_thesis: card C is even
then  ( ( 0 = 0 & card C is even ) or ex k9 being Element of NAT st
( 0 = k9 + 1 & ex C9 being Chain of (k9 + 1),G st
( C9 = C & del C9 = 0_ (k9,G) ) ) ) by Def14;
hence  card C is even ; ::_thesis: verum
end;
thus  ( card C is even implies C is Cycle of 0 ,G ) by Def14; ::_thesis: verum
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C be Cycle of k + 1,G;
redefine func del C equals :: CHAIN_1:def 15
 0_ (k,G);
compatibility
 for b1 being Chain of k,G holds
( b1 = del C iff b1 = 0_ (k,G) ) by Th64;
end;

:: deftheorem  defines del CHAIN_1:def_15_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for C being Cycle of k + 1,G holds del C = 0_ (k,G);

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
:: original: 0_
redefine func 0_ (k,G) ->  Cycle of k,G;
coherence
 0_ (k,G) is Cycle of k,G
proof
percases ( k = 0 or ex k9 being Nat st k = k9 + 1 ) by NAT_1:6;
supposeA1: k = 0 ; ::_thesis: 0_ (k,G) is Cycle of k,G
 card {} = 2 * 0 ;
hence  0_ (k,G) is Cycle of k,G by A1, Def14; ::_thesis: verum
end;
suppose ex k9 being Nat st k = k9 + 1 ; ::_thesis: 0_ (k,G) is Cycle of k,G
then consider k9 being Nat such that
A2: k = k9 + 1 ;
reconsider k9 = k9 as Element of NAT by ORDINAL1:def_12;
 del (0_ ((k9 + 1),G)) = 0_ (k9,G) by Th59;
hence  0_ (k,G) is Cycle of k,G by A2, Def14; ::_thesis: verum
end;
end;
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
:: original: Omega
redefine func Omega G ->  Cycle of d,G;
coherence
 Omega G is Cycle of d,G
proof
consider d9 being Nat such that
A1: d = d9 + 1 by NAT_1:6;
reconsider d9 = d9 as Element of NAT by ORDINAL1:def_12;
reconsider G = G as Grating of d9 + 1 by A1;
 del (Omega G) = 0_ (d9,G) by Th60;
hence  Omega G is Cycle of d,G by A1, Def14; ::_thesis: verum
end;
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C1, C2 be Cycle of k,G;
:: original: +
redefine funcC1 + C2 ->  Cycle of k,G;
coherence
 + is Cycle of k,G
proof
percases ( k = 0 or ex k9 being Nat st k = k9 + 1 ) by NAT_1:6;
supposeA1: k = 0 ; ::_thesis: + is Cycle of k,G
then A2: card C1 is even by Th66;
 card C2 is even by A1, Th66;
then  card (C1 + C2) is even by A2, Th8;
hence  C1 + C2 is Cycle of k,G by A1, Th66; ::_thesis: verum
end;
suppose ex k9 being Nat st k = k9 + 1 ; ::_thesis: + is Cycle of k,G
then consider k9 being Nat such that
A3: k = k9 + 1 ;
reconsider k9 = k9 as Element of NAT by ORDINAL1:def_12;
reconsider C1 = C1, C2 = C2 as Cycle of k9 + 1,G by A3;
A4: del C1 = 0_ (k9,G) ;
 del C2 = 0_ (k9,G) ;
then  del (C1 + C2) = (0_ (k9,G)) + (0_ (k9,G)) by A4, Th61
.= 0_ (k9,G) ;
hence  + is Cycle of k,G by A3, Th64; ::_thesis: verum
end;
end;
end;
end;

theorem :: CHAIN_1:67
for d being non zero Element of NAT
for G being Grating of d
for C being Cycle of d,G holds C ` is Cycle of d,G
proof
let d be non zero Element of NAT ; ::_thesis: for G being Grating of d
for C being Cycle of d,G holds C ` is Cycle of d,G
let G be Grating of d; ::_thesis: for C being Cycle of d,G holds C ` is Cycle of d,G
let C be Cycle of d,G; ::_thesis: C ` is Cycle of d,G
consider d9 being Nat such that
A1: d = d9 + 1 by NAT_1:6;
reconsider d9 = d9 as Element of NAT by ORDINAL1:def_12;
reconsider G = G as Grating of d9 + 1 by A1;
reconsider C = C as Cycle of d9 + 1,G by A1;
 del C = 0_ (d9,G) ;
then  del (C `) = 0_ (d9,G) by Th62;
hence  C ` is Cycle of d,G by A1, Th64; ::_thesis: verum
end;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
let C be Chain of (k + 1),G;
:: original: del
redefine func del C ->  Cycle of k,G;
coherence
 del C is Cycle of k,G
proof
percases ( k = 0 or ex k9 being Nat st k = k9 + 1 ) by NAT_1:6;
supposeA1: k = 0 ; ::_thesis: del C is Cycle of k,G
defpred S1[ Chain of (k + 1),G] means  del $1 is Cycle of k,G;
 del (0_ ((k + 1),G)) = 0_ (k,G) ;
then A2: S1[ 0_ ((k + 1),G)] ;
now__::_thesis:_for_B_being_Cell_of_(0_+_1),G_st_B_in_C_holds_
del_{B}_is_Cycle_of_0_,G
let B be Cell of (0 + 1),G; ::_thesis: ( B in C implies del {B} is Cycle of 0 ,G )
assume  B in C ; ::_thesis: del {B} is Cycle of 0 ,G
 card (del {B}) = 2 * 1 by Th55;
hence  del {B} is Cycle of 0 ,G by Def14; ::_thesis: verum
end;
then A3: for A being Cell of (k + 1),G st A in C holds
S1[{A}] by A1;
A4: for C1, C2 being Chain of (k + 1),G st C1 c= C & C2 c= C & S1[C1] & S1[C2] holds
S1[C1 + C2]
proof
let C1, C2 be Chain of (k + 1),G; ::_thesis: ( C1 c= C & C2 c= C & S1[C1] & S1[C2] implies S1[C1 + C2] )
assume that
 C1 c= C and
 C2 c= C and
A5: S1[C1] and
A6: S1[C2] ; ::_thesis: S1[C1 + C2]
reconsider C19 = del C1, C29 = del C2 as Cycle of k,G by A5, A6;
 del (C1 + C2) = C19 + C29 by Th61;
hence  S1[C1 + C2] ; ::_thesis: verum
end;
thus  S1[C] from CHAIN_1:sch_4(A2, A3, A4); ::_thesis: verum
end;
suppose ex k9 being Nat st k = k9 + 1 ; ::_thesis: del C is Cycle of k,G
then consider k9 being Nat such that
A7: k = k9 + 1 ;
reconsider k9 = k9 as Element of NAT by ORDINAL1:def_12;
reconsider C = C as Chain of ((k9 + 1) + 1),G by A7;
 del (del C) = 0_ (k9,G) by Th63;
hence  del C is Cycle of k,G by A7, Th64; ::_thesis: verum
end;
end;
end;
end;

begin

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
func Chains (k,G) ->  strict AbGroup means :Def16: :: CHAIN_1:def 16
( the carrier of it = bool (cells (k,G)) & 0. it = 0_ (k,G) & ( for A, B being Element of it
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) );
existence
 ex b1 being strict AbGroup st
( the carrier of b1 = bool (cells (k,G)) & 0. b1 = 0_ (k,G) & ( for A, B being Element of b1
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) )
proof
deffunc H2( Chain of k,G, Chain of k,G) ->  Chain of k,G = $1 + $2;
consider op being BinOp of (bool (cells (k,G))) such that
A1: for A, B being Chain of k,G holds op . (A,B) = H2(A,B) from BINOP_1:sch_4();
set ch = addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #);
A2: addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) is add-associative
proof
let A, B, C be Element of addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #); :: according to RLVECT_1:def_3 ::_thesis: (A + B) + C = A + (B + C)
reconsider A9 = A, B9 = B, C9 = C as Chain of k,G ;
thus (A + B) + C = op . ((A9 + B9),C) by A1
.= (A9 + B9) + C9 by A1
.= A9 + (B9 + C9) by XBOOLE_1:91
.= op . (A,(B9 + C9)) by A1
.= A + (B + C) by A1 ; ::_thesis: verum
end;
A3: addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) is right_zeroed
proof
let A be Element of addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #); :: according to RLVECT_1:def_4 ::_thesis: A + (0. addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #)) = A
reconsider A9 = A as Chain of k,G ;
thus A + (0. addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #)) = A9 + (0_ (k,G)) by A1
.= A ; ::_thesis: verum
end;
A4: addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) is right_complementable
proof
let A be Element of addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #); :: according to ALGSTR_0:def_16 ::_thesis: A is right_complementable
reconsider A9 = A as Chain of k,G ;
take  A ; :: according to ALGSTR_0:def_11 ::_thesis: A + A = 0. addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #)
thus A + A = A9 + A9 by A1
.= 0. addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) by XBOOLE_1:92 ; ::_thesis: verum
end;
 addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) is Abelian
proof
let A, B be Element of addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #); :: according to RLVECT_1:def_2 ::_thesis: A + B = B + A
reconsider A9 = A, B9 = B as Chain of k,G ;
thus A + B = A9 + B9 by A1
.= B + A by A1 ; ::_thesis: verum
end;
then reconsider ch = addLoopStr(# (bool (cells (k,G))),op,(0_ (k,G)) #) as strict AbGroup by A2, A3, A4;
take  ch ; ::_thesis: ( the carrier of ch = bool (cells (k,G)) & 0. ch = 0_ (k,G) & ( for A, B being Element of ch
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) )
thus  the carrier of ch = bool (cells (k,G)) ; ::_thesis: ( 0. ch = 0_ (k,G) & ( for A, B being Element of ch
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) )
thus  0. ch = 0_ (k,G) ; ::_thesis: for A, B being Element of ch
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9
let A, B be Element of ch; ::_thesis: for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9
let A9, B9 be Chain of k,G; ::_thesis: ( A = A9 & B = B9 implies A + B = A9 + B9 )
assume that
A5: A = A9 and
A6: B = B9 ; ::_thesis: A + B = A9 + B9
thus  A + B = A9 + B9 by A1, A5, A6; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict AbGroup st the carrier of b1 = bool (cells (k,G)) & 0. b1 = 0_ (k,G) & ( for A, B being Element of b1
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) & the carrier of b2 = bool (cells (k,G)) & 0. b2 = 0_ (k,G) & ( for A, B being Element of b2
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) holds
b1 = b2
proof
let C1, C2 be strict AbGroup; ::_thesis: ( the carrier of C1 = bool (cells (k,G)) & 0. C1 = 0_ (k,G) & ( for A, B being Element of C1
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) & the carrier of C2 = bool (cells (k,G)) & 0. C2 = 0_ (k,G) & ( for A, B being Element of C2
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) implies C1 = C2 )
assume that
A7: the carrier of C1 = bool (cells (k,G)) and
A8: 0. C1 = 0_ (k,G) and
A9: for A, B being Element of C1
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 and
A10: the carrier of C2 = bool (cells (k,G)) and
A11: 0. C2 = 0_ (k,G) and
A12: for A, B being Element of C2
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ; ::_thesis: C1 = C2
set X = [:(bool (cells (k,G))),(bool (cells (k,G))):];
reconsider op1 = the addF of C1, op2 = the addF of C2 as Function of [:(bool (cells (k,G))),(bool (cells (k,G))):],(bool (cells (k,G))) by A7, A10;
now__::_thesis:_for_AB_being_Element_of_[:(bool_(cells_(k,G))),(bool_(cells_(k,G))):]_holds_op1_._AB_=_op2_._AB
let AB be Element of [:(bool (cells (k,G))),(bool (cells (k,G))):]; ::_thesis: op1 . AB = op2 . AB
consider A9, B9 being Chain of k,G such that
A13: AB = [A9,B9] by DOMAIN_1:1;
reconsider A1 = A9, B1 = B9 as Element of C1 by A7;
reconsider A2 = A9, B2 = B9 as Element of C2 by A10;
thus op1 . AB = A1 + B1 by A13
.= A9 + B9 by A9
.= A2 + B2 by A12
.= op2 . AB by A13 ; ::_thesis: verum
end;
hence  C1 = C2 by A7, A8, A10, A11, FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def16 defines Chains CHAIN_1:def_16_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for b4 being strict AbGroup holds
( b4 = Chains (k,G) iff ( the carrier of b4 = bool (cells (k,G)) & 0. b4 = 0_ (k,G) & ( for A, B being Element of b4
for A9, B9 being Chain of k,G st A = A9 & B = B9 holds
A + B = A9 + B9 ) ) );

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
mode GrChain of k,G is Element of (Chains (k,G));
end;

theorem :: CHAIN_1:68
for k being Element of NAT
for d being non zero Element of NAT
for G being Grating of d
for x being set holds
( x is Chain of k,G iff x is GrChain of k,G ) by Def16;

definition
let d be non zero Element of NAT ;
let G be Grating of d;
let k be Element of NAT ;
func del (k,G) ->  Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)) means :: CHAIN_1:def 17
 for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
it . A = del A9;
existence
 ex b1 being Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)) st
for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
b1 . A = del A9
proof
deffunc H2( Subset of (cells ((k + 1),G))) ->  Cycle of k,G = del $1;
consider f being Function of (bool (cells ((k + 1),G))),(bool (cells (k,G))) such that
A1: for A being Subset of (cells ((k + 1),G)) holds f . A = H2(A) from FUNCT_2:sch_4();
A2: the carrier of (Chains ((k + 1),G)) = bool (cells ((k + 1),G)) by Def16;
 the carrier of (Chains (k,G)) = bool (cells (k,G)) by Def16;
then reconsider f9 = f as Function of (Chains ((k + 1),G)),(Chains (k,G)) by A2;
now__::_thesis:_for_A,_B_being_Element_of_(Chains_((k_+_1),G))_holds_f_._(A_+_B)_=_(f9_._A)_+_(f9_._B)
let A, B be Element of (Chains ((k + 1),G)); ::_thesis: f . (A + B) = (f9 . A) + (f9 . B)
reconsider A9 = A, B9 = B as Chain of (k + 1),G by Def16;
thus f . (A + B) = f . (A9 + B9) by Def16
.= del (A9 + B9) by A1
.= (del A9) + (del B9) by Th61
.= (del A9) + (f . B9) by A1
.= (f . A9) + (f . B9) by A1
.= (f9 . A) + (f9 . B) by Def16 ; ::_thesis: verum
end;
then  f9 is additive by VECTSP_1:def_20;
then reconsider f9 = f9 as Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)) ;
take  f9 ; ::_thesis: for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
f9 . A = del A9
thus  for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
f9 . A = del A9 by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)) st ( for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
b1 . A = del A9 ) & ( for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
b2 . A = del A9 ) holds
b1 = b2
proof
let f, g be Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)); ::_thesis: ( ( for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
f . A = del A9 ) & ( for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
g . A = del A9 ) implies f = g )
assume A3: for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
f . A = del A9 ; ::_thesis: ( ex A being Element of (Chains ((k + 1),G)) ex A9 being Chain of (k + 1),G st
( A = A9 & not g . A = del A9 ) or f = g )
assume A4: for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
g . A = del A9 ; ::_thesis: f = g
now__::_thesis:_for_A_being_Element_of_(Chains_((k_+_1),G))_holds_f_._A_=_g_._A
let A be Element of (Chains ((k + 1),G)); ::_thesis: f . A = g . A
reconsider A9 = A as Element of (Chains ((k + 1),G)) ;
reconsider A99 = A as Chain of (k + 1),G by Def16;
f . A9 = del A99 by A3
.= g . A9 by A4 ;
hence  f . A = g . A ; ::_thesis: verum
end;
hence  f = g by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem  defines del CHAIN_1:def_17_:_
 for d being non zero Element of NAT
for G being Grating of d
for k being Element of NAT
for b4 being Homomorphism of (Chains ((k + 1),G)),(Chains (k,G)) holds
( b4 = del (k,G) iff for A being Element of (Chains ((k + 1),G))
for A9 being Chain of (k + 1),G st A = A9 holds
b4 . A = del A9 );